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TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 365, Number 9, September 2013, Pages 4729–4809S 0002-9947(2013)05832-1Article electronically published on February 27, 2013

WEIGHTED LOCAL ORLICZ-HARDY SPACES ON DOMAINS

AND THEIR APPLICATIONS IN INHOMOGENEOUS

DIRICHLET AND NEUMANN PROBLEMS

JUN CAO, DER-CHEN CHANG, DACHUN YANG, AND SIBEI YANG

Abstract. Let Ω be either Rn or a strongly Lipschitz domain of Rn, andω ∈ A∞(Rn) (the class of Muckenhoupt weights). Let L be a second-orderdivergence form elliptic operator on L2(Ω) with the Dirichlet or Neumannboundary condition, and assume that the heat semigroup generated by L hasthe Gaussian property (G1) with the regularity of their kernels measured byμ ∈ (0, 1]. Let Φ be a continuous, strictly increasing, subadditive, positive

and concave function on (0,∞) of critical lower type index p−Φ ∈ (0, 1]. Inthis paper, the authors first introduce the “geometrical” weighted local Orlicz-Hardy spaces hΦ

ω, r(Ω) and hΦω, z(Ω) via the weighted local Orlicz-Hardy spaces

hΦω (Rn), and obtain their two equivalent characterizations in terms of the

nontangential maximal function and the Lusin area function associated withthe heat semigroup generated by L when p−Φ ∈ (n/(n + μ), 1]. Second, the

authors furthermore establish three equivalent characterizations of hΦω, r(Ω) in

terms of the grand maximal function, the radial maximal function and theatomic decomposition when the complement of Ω is unbounded and p−Φ ∈(0, 1]. Third, as applications, the authors prove that the operators ∇2GD arebounded from hΦ

ω, r(Ω) to the weighted Orlicz space LΦω (Ω), and from hΦ

ω, r(Ω)

to itself when Ω is a bounded semiconvex domain in Rn and p−Φ ∈ ( nn+1

, 1], and

the operators ∇2GN are bounded from hΦω, z(Ω) to LΦ

ω (Ω), and from hΦω, z(Ω)

to hΦω, r(Ω) when Ω is a bounded convex domain in Rn and p−Φ ∈ ( n

n+1, 1],

where GD and GN denote, respectively, the Dirichlet Green operator and theNeumann Green operator.

Contents

1. Introduction 47302. Preliminaries 47412.1. Some properties of the weight class A∞(Rn) 47412.2. The divergence form elliptic operator L 47432.3. Orlicz functions 4745

Received by the editors July 25, 2011.2010 Mathematics Subject Classification. Primary 42B35; Secondary 42B30, 42B20, 42B25,

35J25, 42B37, 47B38, 46E30.Key words and phrases. Strongly Lipschitz domain, semiconvex domain, convex domain, di-

vergence form elliptic operator, Dirichlet Green operator, Neumann Green operator, Dirichletboundary condition, Neumann boundary condition, weight, local Orlicz-Hardy space, Gaussianproperty, nontangential maximal function, Lusin area function, atom.

The second author was partially supported by an NSF grant DMS-1203845 and a Hong KongRGC competitive earmarked research grant #601410.

The third (corresponding) author was supported by the National Natural Science Foundationof China (Grant No. 11171027) and the Specialized Research Fund for the Doctoral Program ofHigher Education of China (Grant No. 20120003110003).

c©2013 American Mathematical Society

4729

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

4730 JUN CAO, DER-CHEN CHANG, DACHUN YANG, AND SIBEI YANG

3. Proof of Theorem 1.4 47464. Proof of Theorem 1.7 47815. Proof of Theorem 1.8 47896. Proof of Theorem 1.9 4798Acknowledgements 4805References 4805

1. Introduction

The theory of Hardy spaces on the n-dimensional Euclidean space Rn was origi-nally initiated by Stein and Weiss in [77]. Later, Fefferman and Stein [30] systemat-ically developed a real-variable theory for the Hardy spaces Hp(Rn) with p ∈ (0, 1],which are designed to behave well under the Calderon-Zygmund operators; see, forexample, [19, 67, 75, 76]. In particular, they respect translations, rotations, anddilations. However, there are two shortcomings of the spaces Hp(Rn). It is knownthat the spaces Hp(Rn) are not closed under compositions with diffeomorphismsnor under multiplication by smooth functions with compact support. In order toovercome these issues, Goldberg [37] developed the theory of the local Hardy spaceshp(Rn) with p ∈ (0, 1], which plays an important role in partial differential equa-tions and harmonic analysis; see, for example, [12, 37, 74, 78] and their references.In particular, one may prove that pseudo-differential operators of order zero arebounded on the spaces hp(Rn) with p ∈ (0, 1]; see [37] (also [79, 80]). In [12],Bui studied the weighted version hp

ω(Rn) of the local Hardy space hp(Rn) with

ω ∈ A∞(Rn), where and in what follows, Aq(Rn) for q ∈ [1,∞] denotes the class

of Muckenhoupt weights ; see, for example, [33] for their definitions and properties.Rychkov [74] introduced a class of local weights, denoted by Aloc

∞ (Rn), and stud-ied the weighted Besov-Lipschitz spaces and Triebel-Lizorkin spaces with weightsbelonging to Aloc

∞ (Rn), which contains A∞(Rn) weights as special cases. In par-ticular, Rychkov [74] generalized some of the results of Bui [12] on weighted localHardy spaces hp

ω(Rn) to Aloc

∞ (Rn) weights. Very recently, Tang [78] established theweighted atomic decomposition characterization of the weighted local Hardy spacehpω(R

n) with ω ∈ Aloc∞ (Rn) via the local grand maximal function.

On the other hand, as a generalization of Lp(Rn), the Orlicz space was introducedby Birnbaum-Orlicz in [11] and Orlicz in [68]. Since then, the theory of the Orliczspaces themselves has been well developed and these spaces have been widely usedin probability, statistics, potential theory, and partial differential equations, as wellas harmonic analysis and some other fields of mathematics; see, for example, [13,45, 59, 70, 71]. Moreover, Orlicz-Hardy spaces are also suitable substitutions of theOrlicz spaces in the study of boundedness of operators; see, for example, [46, 48,49, 51, 82]. Recall that Orlicz-Hardy spaces and their dual spaces were studied byJanson [46] on Rn and Viviani [82] on spaces of homogeneous type in the sense ofCoifman and Weiss [21] (see also [22]). Let Φ be a continuous, strictly increasing,subadditive, positive and concave function on (0,∞) of strictly critical lower typeindex pΦ ∈ (0, 1] (see (2.13) below for the definition of pΦ). Based on the worksof Rychkov [74] and Tang [78], the weighted local Orlicz-Hardy spaces hΦ

ω (Rn)

with ω ∈ Aloc∞ (Rn) were introduced and studied in [86]. We point out that the


WEIGHTED LOCAL ORLICZ-HARDY SPACES ON DOMAINS 4731

assumptions on pΦ in [86] can be relaxed into the same assumptions on p−Φ ; see

(2.12) below for the definition of p−Φ and also Remark 2.7 below.As we mentioned at the beginning, Hardy spaces Hp(Rn) are essentially related

to the second-order elliptic operator with constant coefficients,

L :=n∑

j,k=1

ajk∂2

∂x2i

,

where {ajk}nj,k=1 are constants and (ajk)n×n > 0. In recent years, the researchof the real-variable theory of various function spaces associated with different dif-ferential operators has inspired great interests; see, for example, [6, 7, 28, 29, 38,39, 40, 41, 50, 85]. Moreover, Orlicz-Hardy spaces associated with some differen-tial operators and their dual spaces were introduced and studied in [48, 49, 51].In particular, the local Hardy space h1

L(Rn), associated with a linear operator L

in L2(Rn) which generates an analytic semigroup with kernels satisfying an upperbound of Poisson type, was also studied in [52].

Next, it is natural to develop a theory of Hardy spaces on domains of Rn; see,for example, [8, 16, 17, 18, 27, 43, 44, 66, 81]. As we may expect, there are severalways to define Hardy spaces on domains. In particular, the second author of thispaper, Krantz and Stein [18] introduced the Hardy spaces Hp

r (Ω) and Hpz (Ω) on

domains Ω of Rn, respectively, by restricting arbitrary elements of Hp(Rn) to Ω,and restricting elements of Hp(Rn) which are zero outside Ω to Ω, where and inwhat follows, Ω denotes the closure of Ω in Rn. For these Hardy spaces, atomicdecompositions have been obtained in [18] when Ω is a special Lipschitz domainor a bounded Lipschitz domain of Rn. The second author of this paper, Krantzand Stein [18] also introduced the local Hardy spaces hp

r(Ω) and hpz(Ω) in a similar

way and obtained atomic decompositions for these local Hardy spaces when Ω isa special Lipschitz domain or a bounded Lipschitz domain of Rn. Furthermore,the dual spaces of these Hardy spaces and local Hardy spaces were studied in [15].Let Ω be a strongly Lipschitz domain and let H1

r (Ω) and H1z (Ω) be defined as in

[18]. Auscher and Russ [8] proved that H1r (Ω) and H1

z (Ω) are characterized bythe nontangential maximal function and the Lusin area function associated with

{e−t√L}t≥0, respectively, under the so-called Dirichlet and Neumann boundary

conditions, where L is a second-order divergence form elliptic operator such thatfor all t ∈ (0,∞), the kernel of e−tL has the Gaussian property (G∞) (see, forexample, [8, Definition 3] or Definition 2.4 below). Let Φ be a continuous, strictlyincreasing, subadditive, positive function on (0,∞) of strictly critical lower typeindex

pΦ ∈(

n

n+ μ, 1

].

The Orlicz-Hardy spaces HΦr (Ω) and HΦ

z (Ω) were first introduced, respectively, in[87] and [88]. Similar to [86], we point out that the assumptions on pΦ in [87, 88]can also be relaxed into the same assumptions on p−Φ ; see Remark 2.7 below.

Let h1r(Ω) and h1

z(Ω) be defined as in [18]. Auscher and Russ [8] also showed thath1r(Ω) and h1

z(Ω) can be characterized by the local nontangential maximal function

and the local Lusin area function associated with {e−t√L}t≥0, respectively, under

the Dirichlet and the Neumann boundary conditions. Here L is a second-orderdivergence form elliptic operator satisfying the Gaussian property (G1).



Let p ∈ (0, 1] and Ω be a proper open subset of Rn. Let Hpr (Ω) be defined

as in [18]. Miyachi [66] obtained three equivalent characterizations of Hpr (Ω) in

terms of the grand maximal function, the radial maximal function and the atomicdecomposition. The dual theory of Hp

r (Ω) was also obtained in [66]. Let Ω be abounded Lipschitz domain of Rn. Wang and Yang [84] introduced the weightedlocal Hardy spaces on Ω by restricting arbitrary elements of hp

ω(Rn) to Ω with

ω ∈ A∞(Rn). They characterized the space hpω, r(Ω) in terms of the grand maximal

function, the radial maximal function and the atomic decomposition. They alsoapplied these characterizations to harmonic functions defined on bounded Lipschitzdomains.

Let Ω be a domain of Rn. In what follows, we denote by W 1, 2(Ω) the usualSobolev space on Ω equipped with the norm{

‖f‖2L2(Ω) + ‖∇f‖2L2(Ω)

}1/2

,

where ∇f denotes the distributional gradient of f . In what follows, W 1, 20 (Ω) stands

for the closure of C∞c (Ω) in W 1, 2(Ω), where C∞

c (Ω) denotes the set of all C∞

functions on Ω with compact support.Now we recall the inhomogeneous Dirichlet problem and Neumann problem on

bounded domains of Rn. Given an open, bounded subset Ω of Rn and f ∈ C∞(Ω),

we denote by GD(f) the unique solution in W 1, 20 (Ω) of the inhomogeneous Dirichlet

problem

(1.1)

{Δu = f, in Ω,

u = 0, on ∂Ω,

and refer to GD as the Dirichlet Green operator. Given an open, bounded subsetΩ of Rn and f ∈ C∞(Ω), we denote by GN (f) the unique solution in W 1, 2(Ω) ofthe inhomogeneous Neumann problem

(1.2)

{Δu = f, in Ω,

∂νu = 0, on ∂Ω,

where it is assumed that∫Ωf(x) dx = 0 and the solution is normalized by requiring

that∫Ωu(x) dx = 0, ν(x) denotes the unit outward normal to ∂Ω at x ∈ ∂Ω, and

∂ν := ∇ · ν stands for the normal derivative.Let Ω be a bounded smooth domain in Rn. The regularity of the operators

GD and GN on Lp(Ω) spaces, for p ∈ (1,∞), is well known; see, for example,[4, 57]. More results, including extensions to local Hardy spaces hp

r(Ω) or hpz(Ω) for

p ∈ (0, 1], were obtained in [16, 17, 18]. A natural question is to study the regularityof these Green operators on Lp spaces for p ∈ (1,∞) on a bounded domain Ω in Rn

under weaker smoothness hypotheses on the boundary ∂Ω of Ω. Similarly, one mayask whether the Lp(Ω) estimates can be replaced by local Hardy spaces, hp

r(Ω) orhpz(Ω), for p ∈ (0, 1]. We give a brief survey of the progress in this direction via (i)

through (vi) as follows (see also [26]):(i) One early result in this line of work obtained by Kadlec [54] is that the

mappings

(1.3) f �→ ∂2GD(f)

∂xi∂xj, i, j ∈ {1, · · · , n},



are well defined and bounded on L2(Ω) whenever Ω is a bounded convex domainin Rn; see also [1, 35].

(ii) Assume that Ω is a bounded convex domain in Rn. The mappings in (1.3)were shown to be of weak type (1, 1), independently, by Dahlberg et al. [25] andFromm [31], and to be bounded on a suitable Hardy space by Adolfsson [2]. Byinterpolation, these mappings are bounded on Lp(Ω) for p ∈ (1, 2).

(iii) The L2-boundedness of the mappings,

(1.4) f �→ ∂2GN (f)

∂xi∂xj, i, j ∈ {1, · · · , n},

has been known since the mid 1970s (see, for example, [35]) when Ω is a boundedconvex domain in Rn, but the optimal Lp(Ω) estimates, valid in the range p ∈ (1, 2],have only been proved in 1994 by Adolfsson and Jerison [3]. Their method was toobtain an endpoint estimate for atoms in a suitable Hardy space H1(Ω), and thento use interpolation with the L2(Ω) results.

(iv) For p ∈ (0, 1], the regularity of the Green operators GD and GN on scales oflocal Hardy spaces, hp

r(Ω) or hpz(Ω), have been studied by Mayboroda and Mitrea

[60, 61] when Ω is a bounded Lipschitz domain in Rn, and the results were formu-lated in terms of a pair of Hardy spaces, hp

r(Ω) and hpz(Ω), for the range p ∈ ( n

n+ε , 1)

for some ε ∈ (0, 1]. Recently, Mitrea et al. [63] studied the mapping properties ofthe mappings in (1.3) on Besov and Triebel-Lizorkin spaces in a bounded Lipschitzdomain satisfying a uniform exterior ball condition. In particular, the mappings in(1.3) are bounded from hp

r(Ω) into itself when p ∈ ( nn+1 , 1].

(v) Very recently, Mitrea et al. [64] proved that these mappings in (1.3) and(1.4) are bounded on L2(Ω) when Ω is a bounded semiconvex domain in Rn (whichcontains bounded convex domains). Moreover, Duong et al. [26] obtained theboundedness of these mappings in (1.3) from the local Hardy spaces hp

ΔD(Ω) to

Lp(Ω) for the range p ∈ (0, 1] and proved that these mappings are also of weaktype (1, 1), when Ω is a bounded simply connected semiconvex domain in Rn. Byinterpolation, these mappings are bounded on Lp(Ω) for all p ∈ (1, 2). In [26],Duong et al. also showed that these mappings in (1.4) are bounded from thelocal Hardy spaces hp

ΔN(Ω) to Lp(Ω) for the range p ∈ (0, 1] and are of weak type

(1, 1), when Ω is a bounded simply connected semiconvex domain in Rn. Hence,by interpolation again, these mappings are bounded on Lp(Ω) for p ∈ (1, 2). Basedon these results, Duong et al. [26] proved that the mappings in (1.3) are boundedon the local Hardy space hp

r(Ω), for the range p ∈ ( nn+1 , 1], when Ω is a bounded

simply connected semiconvex domain in Rn; and the mappings in (1.4) are boundedfrom the space hp

z(Ω) to the space hpr(Ω) when p ∈ ( n

n+1 , 1] and Ω is a boundedsimply connected convex domain in Rn.

(vi) In relation to (ii) and (iii) above, it should be mentioned that the aforemen-tioned Lp(Ω) boundedness of the mappings in (1.3) and (1.4) may fail in the classof Lipschitz domains Ω for any p ∈ (1,∞) and in the class of convex domains Ωfor any p ∈ (2,∞) (see [2, 3, 24, 47] for counterexamples; recall that every convexdomain is a Lipschitz domain).

Let Ω be either Rn or a strongly Lipschitz domain of Rn, and ω ∈ A∞(Rn). LetL be a second-order divergence form elliptic operator on L2(Ω) with the Dirichletor Neumann boundary condition, and assume that the heat semigroup generatedby L has the Gaussian property (G1) with the regularity of their kernels measured



by μ ∈ (0, 1]. Let Φ be a continuous, strictly increasing, subadditive, positiveand concave function on (0,∞) of critical lower type index p−Φ ∈ (0, 1]. A typicalexample of such functions is

Φ(t) := tp for all t ∈ (0,∞) and p ∈ (0, 1].

More examples are given in Section 2.3 below. First, motivated by [8, 16, 18, 49,51, 86], we introduce the weighted local Orlicz-Hardy spaces hΦ

ω, r(Ω) and hΦω, z(Ω),

respectively, by restricting arbitrary elements of hΦω (R

n) to Ω or by restrictingelements of hΦ

ω (Rn), which are zero outside Ω, to Ω, where hΦ

ω (Rn) denotes the

weighted local Orlicz-Hardy space introduced in [86]. Then we establish the atomicdecompositions of these spaces by means of the Lusin area function associated with{e−tL}t≥0. Applying this, we obtain two equivalent characterizations of hΦ

ω, r(Ω)

and hΦω, z(Ω), respectively, in terms of the nontangential maximal function and

the Lusin area function associated with {e−tL}t≥0. Second, motivated by [66],we furthermore establish three equivalent characterizations of the spaces hΦ

ω, r(Ω),respectively, in terms of the grand maximal function, the radial maximal functionand the atomic decomposition, when the complement of Ω is unbounded and p−Φ ∈(0, 1]. Third, as applications, we prove that the operators ∇2GD are bounded fromhΦω, r(Ω) to the weighted Orlicz space LΦ

ω (Ω) and from hΦω, r(Ω) to itself, when Ω is a

bounded semiconvex domain in Rn and p−Φ ∈ ( nn+1 , 1], and the operators ∇2GN are

bounded from hΦω, z(Ω) to L

Φω (Ω) and from hΦ

ω, z(Ω) to hΦω, r(Ω), when Ω is a bounded

convex domain in Rn and p−Φ ∈ ( nn+1 , 1], where GD and GN denote, respectively,

the Dirichlet Green operator and the Neumann Green operator.To state the main results of this paper, we first recall some necessary notions.

Throughout the whole paper, we always assume that Ω is a strongly Lipschitzdomain of Rn; namely, Ω is a proper open connected set in Rn whose boundary isa finite union of parts of rotated graphs of Lipschitz maps, at most one of theseparts possibly unbounded. It is well known that strongly Lipschitz domains includespecial Lipschitz domains, bounded Lipschitz domains and exterior domains ; see,for example, [8, 10] for their definitions and properties.

Also, throughout the entire paper, for the sake of convenience, we choose thenorm on Rn to be the supremum norm; namely, for any x = (x1, x2, · · · , xn) ∈ Rn,

|x| := max {|x1|, · · · , |xn|} ,for which balls determined by this norm are cubes associated with the usual Eu-clidean norm with sides parallel to the axes.

Remark 1.1. Let Ω be a strongly Lipschitz domain of Rn. Then Ω is a space ofhomogeneous type in the sense of Coifman and Weiss [21]. Furthermore, as a spaceof homogeneous type, the collection of all balls of Ω is given by the set

{Q ∩ Ω : cube Q ⊂ Rn satisfying xQ ∈ Ω and l(Q) ≤ 2 diam (Ω)} ,where xQ denotes the center of Q, l(Q) the sidelength of Q and diam (Ω) thediameter of Ω, namely,

diam (Ω) := sup{|x− y| : x, y ∈ Ω};see, for example, [8].

To introduce the spaces hΦω, r(Ω) and hΦ

ω, z(Ω), we first recall the definition of the

weighted local Orlicz-Hardy space hΦω (R

n) introduced in [86]. Let D(Rn) denote



the space of all infinitely differentiable functions with compact support in Rn en-dowed with the inductive topology, and D′(Rn) its topological dual with the weak-∗topology which is called the space of distributions on Rn. For all f ∈ D′(Rn), letGloc(f) denote its local grand maximal function; see [86, Definition 3.1].

Definition 1.2. Let Φ satisfy Assumption (A) (see Section 2.2 for the definitionof Assumption (A)) and ω ∈ A∞(Rn) (see Definition 2.1 below for the definition ofA∞(Rn)). Define

hΦω (R

n) :=

{f ∈ D′(Rn) :

∫Rn

Φ(Gloc(f)(x)

)ω(x) dx < ∞

}and

‖f‖hΦω(Rn) := inf

{λ ∈ (0,∞) :

∫Rn

Φ

(Gloc(f)(x)

λ

)ω(x) dx ≤ 1

}.

In what follows, let D(Ω) denote the space of all infinitely differentiable functionswith compact support in Ω endowed with the inductive topology, and D′(Ω) itstopological dual with the weak-∗ topology which is called the space of distributionson Ω.

Definition 1.3. Let Φ and ω be as in Definition 1.2, and Ω a subdomain of Rn. Adistribution f on Ω is said to be in the weighted local Orlicz-Hardy space hΦ

ω, r(Ω)

if f is the restriction to Ω of a distribution F in hΦω (R

n); namely,

hΦω, r(Ω) := {f ∈ D′(Ω) : there exists an F ∈ hΦ

ω(Rn) such that F |Ω = f}

= hΦω(R

n)/{F ∈ hΦ

ω (Rn) : F = 0 on Ω

}.

Moreover, for all f ∈ hΦω, r(Ω), the norm of f in hΦ

ω, r(Ω) is defined by

‖f‖hΦω, r(Ω) := inf

{‖F‖hΦ

ω(Ω) : F ∈ hΦω (R

n) and F |Ω = f},

where the infimum is taken over all F ∈ hΦω (R

n) satisfying F = f on Ω.The weighted local Orlicz-Hardy space hΦ

ω, z(Ω) is defined by

hΦω, z(Ω) :=

{f ∈ hΦ

ω (Rn) : f = 0 on (Ω)�

}/{f ∈ hΦ

ω(Rn) : f = 0 on Ω},

where (Ω)� denotes the set Rn \ Ω. Moreover, for any f ∈ hΦω, z(Ω), its norm in

hΦω, z(Ω) is defined by

‖f‖hΦω, z(Ω) := inf

{‖F‖hΦ

ω(Rn) : F ∈ hΦω (R

n), F = 0 on (Ω)� and F |Ω = f}.

Let Ω be either Rn or a strongly Lipschitz domain of Rn. Let Φ satisfy Assump-tion (A), ω ∈ A∞(Rn) and L be a divergence form elliptic operator on L2(Ω) withthe Dirichlet boundary condition (for simplicity, DBC) or the Neumann bound-ary condition (for simplicity, NBC) (see (2.7) below for the definition of L andDefinition 2.3 below for DBC and NBC). Let the spaces hΦ

Nh, ω(Ω), hΦ

˜Sh, ω(Ω) and

hΦSh, ω

(Ω) be respectively as in Definitions 3.1 and 3.3 below. Then one of the mainresults of this paper is as follows.

Theorem 1.4. Let Φ satisfy Assumption (A), ω ∈ A∞(Rn) and L be as in (2.7).Let qω, rω, μ, p

+Φ and p−Φ be respectively as in (2.5), (2.6), (2.9), (2.11) and (2.12).



Let Ω be either Rn or a strongly Lipschitz domain of Rn. Assume that qω, rω, μ,p+Φ and p−Φ satisfy the inequalities qω

p−Φ

< n+μn ,

2qω

p−Φ<

n+ 1

n+

rω − 1

p+Φrω

and rω > 22−qω

, and the semigroup generated by L has the Gaussian property (G1).

(i) If Ω := Rn, then the spaces

hΦω(R

n), hΦNh, ω

(Rn), hΦ˜Sh, ω

(Rn) and hΦSh, ω

(Rn)

coincide with equivalent quasi-norms.(ii) Under DBC, if Ω� is unbounded, then the spaces

hΦω, r(Ω), hΦ

Nh, ω(Ω), hΦ

˜Sh, ω(Ω) and hΦ

Sh, ω(Ω)

coincide with equivalent quasi-norms.(iii) Under NBC, the spaces

hΦω, z(Ω), hΦ

Nh, ω(Ω), hΦ

˜Sh, ω(Ω) and hΦ

Sh, ω(Ω)

coincide with equivalent quasi-norms.

We first point out that the coincidence between hΦω (R

n) and hΦNh, ω

(Rn), or be-

tween hΦω, r(Ω) and hΦ

Nh, ω(Ω), or between hΦ

ω, z(Ω) and hΦNh, ω

(Ω) of Theorem 1.4

when Φ(t) := t for all t ∈ (0,∞) and ω ≡ 1 was already obtained by Auscher andRuss in [8, Theorems 2 and 20].

The proofs of (i) through (iii) of Theorem 1.4 are similar and can be given byfollowing the main approach used by Auscher and Russ in [8] with some subtlemodifications. Here we only give the framework of the proof of Theorem 1.4(ii).The following chains of inequalities give the strategy of the proof of Theorem 1.4(ii).For all f ∈ hΦ

ω, r(Ω) ∩ L2(Ω) and any given R0 ∈ [ 12 ,∞), we have

‖f‖hΦω, r(Ω) �

∥∥∥N loc, 2R0

h (f)∥∥∥LΦ

ω(Ω)(1.5)

�∥∥∥Sloc

h,R0(f)

∥∥∥LΦ

ω(Ω)+ I(f)

�∥∥Sloc

h,R0(f)

∥∥LΦ

ω(Ω)+ I(f) � ‖f‖hΦ

ω, r(Ω),

where the implicit constants are independent of f , and N loc, 2R0

h (f), Sloch,R0

(f) and

Sloch,R0

(f) are respectively as in Definitions 3.1 and 3.3 below, and

I(f) := inf

⎧⎨⎩λ ∈ (0,∞) :∑

˜Qk∈QΩ

ω(Qk ∩ Ω

)Φ

(m

˜Qk∩Ω(|e−R20L(f)|)

λ

)≤ 1

⎫⎬⎭withm

˜Qk∩Ω(|e−R20L(f)|) being the average of |e−R2

0L(f)| on Qk∩Ω (see (3.1) below).

Moreover, see Section 3 for the definitions ofQΩ and Qk. Theorem 1.4(ii) is deducedfrom (1.5) and the arbitrariness of R0 ∈ [ 12 ,∞). The proof of the first inequality

in (1.5) is standard by applying the atomic decomposition of hΦω (R

n) obtainedin [86] and the relation between hΦ

ω, r(Ω) and hΦω (R

n); see Proposition 3.4 below.We prove the second and the third inequalities, respectively, in Propositions 3.8and 3.12 below. We point out that Proposition 3.8 plays an important role inthe proof of Theorem 1.4(ii), and the key step in the proof of Proposition 3.8



is to establish a “good-λ inequality” concerning N loc, 2R0

h (f) and Sloch,R0

(f) (see

Lemma 3.10 below), which is a subtle weighted variant on the local nontangentialmaximal function and the local Lusin-area function of [8, Lemma 9] and even if itsspecial case that ω ≡ 1 also improves [8, Lemma 9] in the sense that [8, Lemma9] presents a “good-λ inequality” on the nontangential maximal function and theaverage of the Lusin-area function, not the Lusin-area function itself (see also [88]).To show the last inequality of (1.5) in Proposition 3.13(i) below, for all f ∈ L2(Ω)satisfying ‖Sloc

h,R0(f)‖LΦ

ω(Ω) < ∞, we establish its atomic decomposition by using a

local Calderon reproducing formula (see (3.39) below) on L2(Ω) associated with L,the atomic decomposition of functions in the tent space on Ω, and the reflectiontechnology related to Lipschitz domains on Rn which was proved by Auscher andRuss in [8, p. 183] and plays a key role in the proof of Theorem 1.4 (see also Lemma3.15 below). But, this reflection technology was not necessary in the proof ofTheorem 1.4(iii) (see also [8]). We point out that the method used in the proofof the last inequality of (1.5) is also quite different from that used in the proof of[8, Proposition 19] for a similar inequality on h1

r(Ω). Since hΦω, r(Ω) when p−Φ < 1

is only a quasi-Banach space, not a Banach space, and the norms of elements inquasi-Banach spaces cannot be achieved by duality, the method used in the proofof [8, Proposition 19] cannot be applied to the current situation; see also [88].

Let p ∈ ( nn+1 , 1]. Recall that bounded convex domains and semiconvex domains

in Rn are strongly Lipschitz domains of Rn. We remark that when L = −Δ with theDirichlet boundary condition, Ω is a bounded semiconvex domain in Rn, ω ≡ 1 andΦ(t) := tp for all t ∈ (0,∞), Theorem 1.4(ii) coincides with [26, Proposition 5.3(ii)];when L = −Δ with the Neumann boundary condition, Ω is a bounded convexdomain in Rn, ω ≡ 1 and Φ(t) := tp for all t ∈ (0,∞), Theorem 1.4(iii) was alsoobtained in [26, Proposition 5.3(i)]. We remark that the approach used in theproofs of [26, Theorem 3.5 and Proposition 5.3] is quite different from that usedin the proof of Theorem 1.4. A key tool used in the proofs of [26, Theorem 3.5and Proposition 5.3] is the atomic characterization closely associated with −Δ ondomains, while in the proof of Theorem 1.4, we fully use the atomic characterizationof hΦ

ω(Rn) in [86].

To state the second main theorem of this paper, we first recall the followingnotions of the radial and the grand maximal functions on Ω.

Definition 1.5. Let Ω be an open subset of Rn. Denote the boundary of Ω by ∂Ω.For all x ∈ Ω, let δ(x) := dist (x, ∂Ω). Let

(1.6) ϕ ∈ D(Rn) with supp (ϕ) ⊂ B(0, 1) and

∫Rn

ϕ(x) dx = 1.

Let c0 ∈ (1,∞). For any f ∈ D′(Ω), the radial maximal function f+ϕ,Ω of f associ-

ated to ϕ on Ω is defined by setting, for all x ∈ Ω,

(1.7) f+ϕ,Ω(x) := sup

0<t<δ(x)/c0

|ϕt ∗ f(x)| ,

where and in what follows, for all t ∈ (0,∞) and x ∈ Rn,

ϕt(x) :=1

tnϕ(xt

).



The grand maximal function f∗Ω of f on Ω is defined by setting, for all x ∈ Ω,

(1.8) f∗Ω(x) := sup

0<t<δ(x)/c0

supψ∈Ft(x)

|〈f, ψ〉|,

where for all x ∈ Ω,

Ft(x) :=

{ψ ∈ D(Rn) : supp (ψ) ⊂ B(x, t),

supy∈Rn

|∂αψ(y)| ≤ t−|α|−n for every α ∈ Zn+

}.

We also need the following notions of local (ρ, q, s)ω-atoms on Ω. Recall thatthe space Lq

ω(Ω), for q ∈ [1,∞] and ω ∈ A∞(Rn), denotes the weighted Lebesguespace endowed with the norm that, for any f ∈ Lq

ω(Ω), when q ∈ [1,∞),

‖f‖Lqω(Ω) :=

{∫Ω

|f(x)|qω(x) dx}1/q

,

and‖f‖L∞

ω (ω) := ‖f‖L∞(Ω).

Definition 1.6. Let Ω be an open subset of Rn and ω ∈ A∞(Rn), qω, p−Φ and ρ

be respectively as in (2.5), (2.12) and (2.14) below. A triplet (ρ, q, s)ω is calledadmissible if q ∈ (qω,∞] and s ∈ Z+ with s ≥ �n( qω

p−Φ

− 1)�. A function a supported

on a cube Q ⊂ Ω is called a type (a) local (ρ, q, s)ω-atom if 4Q ∩ ∂Ω = ∅ withl(Q) < 1,

‖a‖Lqω(Rn) ≤ [ω(Q)]

1q−1[ρ(ω(Q))]−1,

and ∫Rn

a(x)xα dx = 0

for all α ∈ Zn+ with |α| ≤ s.

A function a supported on a cube Q ⊂ Ω is called a type (b) local (ρ, q, s)ω-atomif either l(Q) ≥ 1 or 2Q ∩ ∂Ω = ∅ and 4Q ∩ ∂Ω �= ∅, and


1q−1[ρ(ω(Q))]−1.

Theorem 1.7. Let Φ satisfy Assumption (A), ω ∈ A∞(Rn), Ω be a strongly Lip-

schitz domain of Rn satisfying that Ω� is unbounded, and ϕ be as in (1.6). Thenthe following are equivalent:

(i) f ∈ hΦω, r(Ω);

(ii) f ∈ D′(Ω) and f+ϕ,Ω ∈ LΦ

ω (Ω), where f+ϕ,Ω is as in (1.7);

(iii) f ∈ D′(Ω) and f∗Ω ∈ LΦ

ω (Ω), where f∗Ω is as in (1.8);

(iv) f ∈ D′(Ω) and

f =∑

type (a) atoms

λQaQ +∑

type (b) atoms

μQbQ

in D′(Ω), where (ρ, q, s)ω is an admissible triplet, {aQ}Q is a sequence of type (a)local (ρ, q, s)ω-atoms, {bQ}Q is a sequence of type (b) local (ρ, q, s)ω-atoms and{λQ}Q ∪ {μQ}Q ⊂ C satisfying∑

type (a) atoms

ω(Q)Φ

(|λQ|

ω(Q)ρ(ω(Q))

)



+∑

type (b) atoms

ω(Q)Φ

(|λQ|

ω(Q)ρ(ω(Q))

)< ∞.

Moreover, define

Λ({λQaQ}Q ∪ {μQbQ}Q)

:= inf

{λ ∈ (0,∞) :

∑type (a) atoms

ω(Q)Φ

(|λQ|

λω(Q)ρ(ω(Q))

)

+∑

type (b) atoms

ω(Q)Φ

(|λQ|

λω(Q)ρ(ω(Q))

)≤ 1

⎫⎬⎭and

‖f‖hρ, q, sω (Ω) := inf {Λ ({λQaQ}Q ∪ {μQbQ}Q)} ,

where the infimum is taken over all the decompositions of f as above.Furthermore, for all f ∈ hΦ

ω, r(Ω),

‖f‖hΦω, r(Ω) ∼

∥∥∥f+ϕ,Ω

∥∥∥LΦ

ω(Ω)∼ ‖f∗

Ω‖LΦω(Ω) ∼ ‖f‖hρ, q, s

ω (Ω),

where the implicit constants are independent of f .

The following outline gives the strategy of the proof of Theorem 1.7: (i) ⇐⇒(iv) =⇒ (iii) ⇐⇒ (ii) =⇒ (i). We divide the proof of Theorem 1.7 into the followingfour steps. Step I: (i) ⇐⇒ (iv); Step II: (iii) ⇐⇒ (ii); Step III: (iv) =⇒ (iii); StepIV: (ii) =⇒ (i). In Step I, we prove that (i) implies (iv) by using the atomiccharacterization of hΦ

ω (Rn) and the definition of the space hΦ

ω, r(Ω); moreover, for

any given f ∈ D′(Ω) satisfying (iv) of Theorem 1.7, we construct an F ∈ hΦω (R

n)such that F |Ω = f and

‖F‖hΦω(Rn) � ‖f‖hρ, q, s

ω (Ω)

by using the reflection technology related to Lipschitz domains on Rn from [8](see also Lemma 3.15 below), which completes the proof that (iv) implies (i). InStep II, it is obvious that (iii) implies (ii); by using a useful estimate concerningf+Ω, ϕ and f∗

Ω, which was established by Miyachi [65] (see also (4.5) below), we

prove that (ii) implies (iii). The proof of Step III is standard, which is similar tothat of Proposition 3.4(i) below. In Step IV, for any given f ∈ D′(Ω) satisfyingf+ϕ,Ω ∈ LΦ

ω (Ω), we construct an F ∈ hΦω (R

n) such that F |Ω = f by using two useful

estimates established by Miyachi [66] (see also (4.7) and (4.8) below), which showsthat (ii) implies (i). We remark that except for its own interest, Theorem 1.7 isalso needed to obtain Theorems 1.8 and 1.9 below.

Theorem 1.8. Let Φ, ω, qω, rω and p−Φ be as in Theorem 1.4. Let Ω be a bounded,simply connected, semiconvex domain in Rn, and GD the Dirichlet Green operatorfor the problem (1.1). Then

(i) the operators in (1.3), originally defined on C∞(Ω), can be extended tobounded operators from hΦ

ω, r(Ω) to LΦω (Ω);

(ii) the operators in (1.3), originally defined on C∞(Ω), can be extended tobounded operators from hΦ

ω, r(Ω) to hΦω, r(Ω).



The proof of Theorem 1.8 below follows the approach used in the proofs ofDuong et al. [26, Theorems 4.1 and 5.4]. We prove Theorem 1.8(i) by using theatomic characterization of hΦ

ω, r(Ω) obtained in Theorem 1.7, some useful estimatesabout the integral kernel of the Dirichlet Green operator GD from Fromm [31] andGruter-Widman [36] (see also Lemma 5.2 below), and the Lp(Ω)-boundedness ofthe mappings in (1.3) from Duong et al. [26, Theorem 4.1] (see also Lemma 5.3below). Furthermore, we prove Theorem 1.8(ii) by using Theorem 1.8(i), the radialmaximal function characterization of hΦ

ω, r(Ω) obtained in Theorem 1.7 and thereflection technology related to Lipschitz domains on Rn from [8].

Let p ∈ (n/(n + 1), 1]. We point out that Theorem 1.8(i) when Φ(t) := tp forall t ∈ (0,∞) and ω ≡ 1 was obtained in [26, Theorem 4.1]; and Theorem 1.8(ii)completely covers [26, Theorem 5.4] by taking Φ(t) := tp for all t ∈ (0,∞) andω ≡ 1.

Theorem 1.9. Let Φ, ω, qω, rω and p−Φ be as in Theorem 1.4. Let Ω be a bounded,simply connected, convex domain in Rn. Let GN be the Neumann Green operatorfor the problem (1.2). Then

(i) the operators in (1.4), originally defined on{f ∈ C∞(Ω) :

∫Ω

f(x) dx = 0

},

can be extended to bounded operators from hΦω, z(Ω) to LΦ

ω (Ω);(ii) the operators in (1.4), originally defined on{

f ∈ C∞(Ω) :

∫Ω

f(x) dx = 0

},

can be extended to bounded operators from hΦω, z(Ω) to hΦ

ω, r(Ω).

Because the integral kernel of the Neumann Green operator GN and its gradientsdo not have good enough size estimates which are similar to those in Lemma 5.3,the method used in the proof of Theorem 1.8 is not valid in the proof of Theo-rem 1.9. Motivated by Duong et al. [26, Theorem 3.5 and Proposition 4.11], weestablish a new atomic decomposition characterization of hΦ

ω, z(Ω) via a class ofatoms associated with the operator L, where L denotes the Laplace operator onL2(Ω) with the Neumann boundary condition; see Lemma 6.3 below. By using thisauxiliary result and some integral estimates of the kernel {Kt}t≥0 of the semigroup{e−tL}t≥0 (see Lemma 6.4 below), and the Lp(Ω)-boundedness of the mappings in(1.4) from Duong et al. [26, Theorem 4.2] (see also Lemma 6.5 below), we proveTheorem 1.9(i). Furthermore, from the new atomic decomposition characterizationof hΦ

ω, z(Ω), Theorem 1.9(i) and the radial maximal function characterization of

hΦω, r(Ω) obtained in Theorem 1.7, we deduce Theorem 1.9(ii).Let p ∈ ( n

n+1 , 1]. Theorem 1.9(i) when Φ(t) := tp for all t ∈ (0,∞) and ω ≡ 1

was obtained in [26, Theorem 4.2 and Proposition 5.3(i)]; also Theorem 1.9(ii)completely covers [26, Theorem 5.7] by taking Φ(t) := tp for all t ∈ (0,∞) andω ≡ 1.

By [74], we know that the local weight class Aloc∞ (Rn) satisfies the exponential

growth in the whole space Rn, which implies that when ω ∈ Aloc∞ (Rn) but ω �∈

A∞(Rn), (3.9) and (3.10) in the proof of Proposition 3.4, and (3.20) in the proofof Proposition 3.8 may not be true, respectively, due to the lack of the compact



support of the kernel of e−t2L or the exponential growth of ω. Thus, throughoutthe whole paper, we always assume that ω ∈ A∞(Rn).

The layout of this paper is as follows. In Section 2, we first recall the definition ofthe weight class A∞(Rn) and some of its properties; then we recall some propertiesof the divergence form elliptic operator L on Rn or a strongly Lipschitz domainΩ, and then describe some basic assumptions on L; finally we describe some basicassumptions on Orlicz functions and present some properties of these functions. InSection 3, we give the proof of Theorem 1.4. Section 4 is devoted to the proof ofTheorem 1.7. In Sections 5 and 6, we respectively give the proofs of Theorems 1.8and 1.9.

Finally we make some conventions on notation. Throughout the entire paper, Lalways denotes the second-order divergence form elliptic operator as in (2.7). Wedenote by C a positive constant which is independent of the main parameters, butit may vary from line to line. We also use C(γ, β, · · · ) to denote a positive constantdepending on the indicated parameters γ, β, · · · . The symbol A � B means thatA ≤ CB. If A � B and B � A, then we write A ∼ B. The symbol �s� for s ∈ R

denotes the maximal integer not more than s; Q(x, t) denotes a closed cube in Rn

with center x ∈ Rn and sidelength l(Q) := t and CQ(x, t) := Q(x,Ct). For anygiven normed spaces A and B with the corresponding norms ‖·‖A and ‖·‖B, A ⊂ Bmeans that for all f ∈ A, then f ∈ B and ‖f‖B � ‖f‖A. For any subset G of Rn,

we denote by G� the set Rn \G, and by χG its characteristic function. We also setN := {1, 2, · · · } and Z+ := N ∪ {0}. For any θ := (θ1, . . . , θn) ∈ Zn

+, let

|θ| := θ1 + · · ·+ θn and ∂θx :=

∂|θ|

∂xθ11 · · · ∂xθn

n

.

For any sets E, F ⊂ Rn and z ∈ Rn, let

dist (E,F ) := infx∈E, y∈F

|x− y| and dist (z, E) := infx∈E

|x− z|.

2. Preliminaries

In Subsection 2.1, we first recall some properties of the class of Muckenhouptweights; in Subsection 2.2, we then recall some properties of the divergence formelliptic operator L on Rn or a strongly Lipschitz domain Ω, and describe some basicassumptions on L; in Subsection 2.3, we describe some basic assumptions of Orliczfunctions and then present some properties of these functions.

2.1. Some properties of the weight class A∞(Rn). In this subsection, we firstrecall the definitions and some properties of the class of Muckenhoupt weights andthe reverse Holder class; see, for example, [32, 33, 34].

Definition 2.1. (i) Let p ∈ (1,∞). The weight class Ap(Rn) is defined to be the

set of all nonnegative locally integrable functions ω on Rn such that

(2.1) Ap(ω) := supQ⊂Rn

1

|Q|p∫Q

ω(x) dx

(∫Q

[ω(x)]−p′p dx

) p

p′

< ∞,

where the supremum is taken over all cubes Q ⊂ Rn and 1p + 1

p′ = 1.



When p = 1, the weight class A1(Rn) is defined to be the set of all nonnegative

locally integrable functions ω on Rn such that

(2.2) A1(ω) := supQ⊂Rn

1

|Q|

∫Q

ω(x) dx

(ess supy∈Q

[ω(y)]−1

)< ∞,

where the supremum is taken over all cubes Q ⊂ Rn.(ii) Let r ∈ (1,∞). The reverse Holder class RHr(R

n) is defined to be the setof all nonnegative locally integrable functions ω on Rn such that

RHr(ω) := supQ⊂Rn

(1

|Q|

∫Q

[ω(x)]r dx

)1/r [1

|Q|

∫Q

ω(x) dx

]−1

< ∞,(2.3)

where the supremum is taken over all cubes Q ⊂ Rn.When r = ∞, the reverse Holder class RH∞(Rn) is defined to be the set of all

nonnegative locally integrable functions ω on Rn such that

RH∞(ω) := supQ⊂Rn

[ess supy∈Q

ω(y)

] [1

|Q|

∫Q

ω(x) dx

]−1

< ∞,(2.4)

where the supremum is taken over all cubes Q ⊂ Rn.

In what follows, for a Lebesgue measurable set E ⊂ Rn and a weight ω ∈A∞(Rn), let

ω(E) :=

∫E

ω(x) dx.

Then we recall some properties of the Muckenhoupt weights as follows.

Lemma 2.2. (i) If 1 ≤ p1 < p2 < ∞, then Ap1(Rn) ⊂ Ap2

(Rn).(ii) If ω ∈ Ap(R

n) with p ∈ (1,∞), then there exists q ∈ (1, p) such that ω ∈Aq(R

n).(iii) If ω ∈ Ap(R

n) with p ∈ [1,∞), then there exists a positive constant C suchthat for any cube Q ⊂ Rn and any measurable subset E of Q,

ω(Q)

ω(E)≤ C

[|Q||E|

]p.

(iv) If 1 < p1 ≤ p2 < ∞, then

RH∞(Rn) ⊂ RHp2(Rn) ⊂ RHp1

(Rn).

(v) If ω ∈ RHr(Rn) with r ∈ (1,∞], then there exists a positive constant C such

that for any cube Q ⊂ Rn and any measurable subset E of Q,

ω(E)

ω(Q)≤ C

[|E||Q|

] r−1r

.

(vi) ⋃1≤p<∞

Ap(Rn) =

⋃1<q≤∞

RHq(Rn).

(vii) Let ω ∈ Ap(Rn) with p ∈ (1,∞), R0 ∈ (0,∞) and N loc, R0

h be as in Defini-

tion 3.1 below. Then N loc, R0

h is bounded on Lpω(R

n).



Proof. The statements (i) through (vi) of this lemma and their proofs can be foundin [33, 34]. We omit the details. Now we prove Lemma 2.2(vii). Denote by M theHardy-Littlewood maximal operator on Rn. Let p ∈ (1,∞) and ω ∈ Ap(R

n). From(2.8) below, we deduce that

N loc, R0

h (f) � M(f)

for all f ∈ Lpω(R

n). By this and the well-known fact that M is bounded on Lpω(R

n),

we know that N loc, R0

h is bounded on Lpω(R

n), which completes the proof of Lemma2.2(vii) and hence Lemma 2.2. �

Let

A∞(Rn) :=⋃

1≤p<∞Ap(R

n) =⋃

1<q≤∞RHq(R

n).

For any given ω ∈ A∞(Rn), define the critical indexes qω and rω of ω, respectively,by

(2.5) qω := inf {p ∈ [1,∞) : ω ∈ Ap(Rn)}

and

(2.6) rω := sup {r ∈ (1,∞] : ω ∈ RHr(Rn)} .

Recall that if qω ∈ (1,∞), then ω �∈ Aqω (Rn), and there exists ω �∈ A1(R

n) suchthat qω = 1 (see [53]). Similarly, if rω ∈ (1,∞), then ω �∈ RHrω (R

n), and thereexists ω �∈ RH∞(Rn) such that rω = ∞ (see [23]).

2.2. The divergence form elliptic operator L. In this subsection, we describethe divergence form elliptic operators considered in this paper and the most typicalexample is the Laplace operator on the Lipschitz domain of Rn with the Dirichletboundary condition or the Neumann boundary condition.

For A : Rn → Mn(C) a measurable function, where Mn(C) denotes the set ofall n× n complex-valued matrixes, let

‖A‖∞ := ess supx∈Rn, ξ, η∈Cn,

|ξ|=|η|=1

|A(x)ξ · η|,

where η denotes the conjugate vector of η. For all δ ∈ (0, 1], denote by A(δ) the classof all measurable functions A : Rn → Mn(C) satisfying the ellipticity condition;namely, for all x ∈ Rn and ξ ∈ Cn,

‖A‖∞ ≤ δ−1 and �(A(x)ξ · ξ) ≥ δ|ξ|2,where and in what follows, �(A(x)ξ · ξ) denotes the real part of A(x)ξ · ξ. Denoteby A the union of all A(δ) for δ ∈ (0, 1].

When A ∈ A and V is a closed subspace of W 1, 2(Ω) containing W 1, 20 (Ω), de-

note by L the maximal-accretive operator (see [69, p. 23, Definition 1.46] for thedefinition) on L2(Ω) with largest domain D(L) ⊂ V such that for all f ∈ D(L) andg ∈ V ,

〈Lf, g〉 =∫Ω

A(x)∇f(x) · ∇g(x) dx,

where 〈·, ·〉 denotes the interior product in L2(Ω). In this sense, for all f ∈ D(L),we write

(2.7) Lf := −div(A∇f).



We recall the following Dirichlet and Neumann boundary conditions of L from[8, p. 152].

Definition 2.3. Let Ω be either Rn or a strongly Lipschitz domain of Rn and Lbe as in (2.7). The operator L is said to satisfy the Dirichlet boundary condition

(for simplicity, DBC) if V := W 1, 20 (Ω), and the Neumann boundary condition (for

simplicity, NBC) if V := W 1, 2(Ω).

Recall that if Ω := Rn, then

W 1, 20 (Ω) = W 1, 2(Ω).

Thus, in this case, DBC and NBC are identical.Let L be as in (2.7). Then L generates a semigroup {e−tL}t≥0 of operators

that is analytic (namely, it has a holomorphic extension to a complex half conez �= 0 and |argz| < μ for some μ ∈ (0, π/2)) and contracting on L2(Ω) (namely,for all f ∈ L2(Ω) and t ∈ (0,∞), ‖e−tLf‖L2(Ω) ≤ ‖f‖L2(Ω)); see, for example,[69, Proposition 1.51 and Theorem 1.52] for the details. Also, L has a unique

maximal accretive square root√L such that −

√L generates an analytic and L2(Ω)-

contracting semigroup {Pt}t≥0 with Pt := e−t√L, the Poisson semigroup for L; see,

for example, [55] for the details.Now we recall the Gaussian property of {e−tL}t≥0 on a strongly Lipschitz do-

main; see, for example, [8, Definition 3] and also [9, 10].

Definition 2.4. Let Ω be either Rn or a strongly Lipschitz domain of Rn, and letL be as in (2.7). Let β ∈ (0,∞]. The semigroup generated by L is said to have theGaussian property (Gβ) if the following (i) and (ii) hold:

(i) The kernel of e−tL, denoted by Kt, is a measurable function on Ω × Ω andthere exist positive constants C and α such that for all t ∈ (0, β) and all x, y ∈ Ω,

(2.8) |Kt(x, y)| ≤C

tn/2e−α |x−y|2

t .

(ii) For all x ∈ Ω and t ∈ (0, β), the functions y �→ Kt(x, y) and y �→ Kt(y, x)are Holder continuous in Ω and there exist positive constants C and μ ∈ (0, 1] suchthat for all t ∈ (0, β) and x, y1, y2 ∈ Ω,

(2.9) |Kt(x, y1)−Kt(x, y2)|+ |Kt(y1, x)−Kt(y2, x)| ≤C

tn/2|y1 − y2|μ

tμ/2.

Remark 2.5. (i) The assumption (G∞) is always satisfied if L is the Laplacian ora real operator (under DBC or NBC) on Rn or on Lipschitz domains except underNBC with Ω bounded; see, for example, [10].

(ii) The assumption (G∞) implies that for all β ∈ (0,∞), (Gβ) holds. If β isfinite, by [8, p. 178, Lemma A.1] and the property of semigroups, we know that(Gβ) and (G1) are equivalent.

The following well-known fact is a simple corollary of the analyticity of thesemigroup {e−tL}t≥0. We omit the details.

Lemma 2.6. Let β ∈ (0,∞]. Assume that L has the Gaussian property (Gβ).Then the estimate (2.8) also holds for t∂tKt.



2.3. Orlicz functions. Let Φ be a positive function on R+ := (0,∞). The functionΦ is said to be of upper type p (resp. lower type p) for some p ∈ [0,∞) if there existsa positive constant C such that for all t ∈ [1,∞) (resp. t ∈ (0, 1]) and s ∈ (0,∞),

(2.10) Φ(st) ≤ CtpΦ(s).

Obviously, if Φ is of lower type p for some p ∈ (0,∞), then limt→0+ Φ(t) = 0. So,for the sake of convenience, if it is necessary, we may assume that Φ(0) = 0. If Φ isof both upper type p1 and lower type p0, then Φ is said to be of type (p0, p1). Let

p+Φ := inf{p ∈ (0,∞) : there exists C ∈ (0,∞)(2.11)

such that (2.10) holds for all t ∈ [1,∞) and s ∈ (0,∞)}

and

p−Φ := sup{p ∈ (0,∞) : there exists C ∈ (0,∞)(2.12)

such that (2.10) holds for all t ∈ (0, 1] and s ∈ (0,∞)}.

The function Φ is said to be of strictly lower type p if for all t ∈ (0, 1) ands ∈ (0,∞), Φ(st) ≤ tpΦ(s), and we define

pΦ := sup{p ∈ (0,∞) : Φ(st) ≤ tpΦ(s) holds(2.13)

for all t ∈ (0, 1) and s ∈ (0,∞)}.

It is easy to see that

pΦ ≤ p−Φ ≤ p+Φ

for all Φ. In what follows, pΦ, p−Φ and p+Φ are respectively called the strictly critical

lower type index, the critical lower type index and the critical upper type index ofΦ. Moreover, it was proved in [49, Remark 2.1] that Φ is also of strictly lower typepΦ. In other words, pΦ is attainable.

Throughout the entire paper, we always assume that Φ satisfies the followingassumptions.

Assumption (A). Let Φ be a positive function defined on R+ which is of criti-cal lower type p−Φ ∈ (0, 1]. Also assume that Φ is continuous, strictly increasing,subadditive and concave.

Remark 2.7. We observe that, via the Aoki-Rolewicz theorem in [5, 72], all resultsin [86, 87, 88] are still true if the assumptions on pΦ therein are relaxed into thesame assumptions on p−Φ .

It is worth noticing that p−Φ and p+Φ may not be attainable. For example, forp ∈ (0, 1], if Φ(t) := tp for all t ∈ (0,∞), then Φ satisfies Assumption (A) andpΦ = p−Φ = p+Φ = p; for p ∈ [ 12 , 1], if

Φ(t) := tp/ ln(e+ t)

for all t ∈ (0,∞), then Φ satisfies Assumption (A) and p−Φ = p = p+Φ , p−Φ is not

attainable, but p+Φ is attainable; for p ∈ (0, 12 ], if

Φ(t) := tp ln(e+ t)

for all t ∈ (0,∞), then Φ satisfies Assumption (A) and p−Φ = p = p+Φ , p−Φ is

attainable, but p+Φ is not attainable.



Notice that if Φ satisfies Assumption (A), then Φ(0) = 0 and Φ is obviously of

upper type 1. For any concave and positive function Φ of lower type p, if we set

Φ(t) :=

∫ t

0

Φ(s)

sds

for t ∈ [0,∞), then by [82, Proposition 3.1], Φ is equivalent to Φ; namely, thereexists a positive constant C such that

C−1Φ(t) ≤ Φ(t) ≤ CΦ(t)

for all t ∈ [0,∞). Moreover, Φ is a strictly increasing, concave, subadditive andcontinuous function of lower type p. Notice that all our results are invariant onequivalent functions satisfying Assumption (A). From this, we deduce that all re-sults in this paper with Φ as in Assumption (A) also hold for all concave and

positive functions Φ of the same critical lower type p−Φ as Φ.Since Φ is strictly increasing, we can define the function ρ(t) on R+ by setting,

for all t ∈ (0,∞),

(2.14) ρ(t) :=t−1

Φ−1(t−1),

where Φ−1 is the inverse function of Φ. Then the types of Φ and ρ have the followingrelation: If 0 < p0 ≤ p1 ≤ 1 and Φ is an increasing function, then Φ is of type(p0, p1) if and only if ρ is of type (p−1

1 − 1, p−10 − 1); see [82] for its proof.

3. Proof of Theorem 1.4

In this section, we first introduce the Orlicz-Hardy spaces

hΦNh, ω

(Ω), hΦSh, ω

(Ω) and hΦ˜Sh, ω

(Ω),

and then give the proof of Theorem 1.4.Let Ω be either Rn or a strongly Lipschitz domain of Rn and ω ∈ A∞(Rn).

Recall that for an Orlicz function Φ on (0,∞), a measurable function f on Ω is saidto be in the space LΦ

ω (Ω) if ∫Ω

Φ(|f(x)|)ω(x) dx < ∞.

Moreover, for any f ∈ LΦω (Ω), define its quasi-norm by

‖f‖LΦω(Ω) := inf

{λ ∈ (0,∞) :

∫Ω

Φ

(|f(x)|λ

)ω(x) dx ≤ 1

}.

If p ∈ (0, 1] and Φ(t) := tp for all t ∈ (0,∞), we then denote LΦω (Ω) simply by

Lpω(Ω).

Definition 3.1. Let Φ satisfy Assumption (A), ω ∈ A∞(Rn), Ω be either Rn ora strongly Lipschitz domain of Rn and L be as in (2.7). Let R0 ∈ (0,∞). For allf ∈ L2(Ω) and x ∈ Ω, let

N loc, R0

h (f)(x) := supy∈Ω, t∈(0,R0], |y−x|<t

∣∣∣e−t2L(f)(y)∣∣∣ .



When R0 := 1, denote N loc, R0

h (f) simply by N loch (f). A function f ∈ L2(Ω) is said

to be in hΦNh, ω

(Ω) if N loch (f) ∈ LΦ

ω (Ω); moreover, define

‖f‖hΦNh, ω(Ω) :=

∥∥N loch (f)

∥∥LΦ

ω(Ω)

:= inf

{λ ∈ (0,∞) :

∫Ω

Φ

(N loc

h (f)(x)

λ

)ω(x) dx ≤ 1

}.

The weighted local Orlicz-Hardy space hΦNh, ω

(Ω) is defined to be the completion of

hΦNh, ω

(Ω) in the quasi-norm ‖ · ‖hΦNh, ω(Ω).

Remark 3.2. (i) It is easy to see that ‖ · ‖hΦNh, ω(Ω) is a quasi-norm.

(ii) From the Aoki-Rolewicz theorem in [5, 72], it follows that there exist a

quasi-norm ‖| · ‖| on hΦNh, ω

(Ω) and γ ∈ (0, 1] such that for all f ∈ hΦNh, ω

(Ω),

‖|f‖| ∼ ‖f‖hΦNh, ω(Ω),

and that for any sequence {fj}j ⊂ hΦNh, ω

(Ω),∥∥∥∥∥∥∣∣∣∣∣∣∑j

fj

∥∥∥∥∥∥∣∣∣∣∣∣γ

≤∑j

‖|fj‖|γ .

By the theorem of completion of Yosida [89, p. 56], it follows that (hΦNh, ω

(Ω), ‖| · ‖|)has a completion space (hΦ

Nh, ω(Ω), ‖| · ‖|); namely, for any f ∈ (hΦ

Nh, ω(Ω), ‖| · ‖|),

there exists a Cauchy sequence {fk}∞k=1 ⊂ (hΦNh, ω

(Ω), ‖| · ‖|) such that

limk→∞

‖|fk − f‖| = 0.

Moreover, if {fk}∞k=1 is a Cauchy sequence in (hΦNh, ω

(Ω), ‖| · ‖|), then there exists a

unique f ∈ (hΦNh, ω

(Ω), ‖| · ‖|) such that

limk→∞

‖|fk − f‖| = 0.

Furthermore, by ‖|f‖| ∼ ‖f‖hΦNh, ω(Ω) for all f ∈ hΦ

Nh, ω(Ω), we know that the spaces

(hΦNh, ω

(Ω), ‖·‖hΦNh, ω(Ω)) and (hΦ

Nh, ω(Ω), ‖|·‖|) coincide with equivalent quasi-norms.

In what follows, Q(x, t) denotes the closed cube of Rn centered at x and ofsidelength t with sides parallel to the axes. Similarly, given Q := Q(x, t) and λ ∈(0,∞), we write λQ for the λ-dilated cube, which is the cube with the same centerx and with sidelength λt. Let R0 ∈ (0,∞). For any f ∈ L2(Ω) and x ∈ Ω, the

Lusin area functions Sloch,R0

(f)(x) and Sloch,R0

(f)(x) associated with {e−t2L}t≥0 arerespectively defined by

Sloch,R0

(f)(x) :=

[∫ΓR0(x)

∣∣∣t2Le−t2L(f)(y)∣∣∣2 dy dt

t|Q(x, t) ∩ Ω|

] 12

and

Sloch,R0

(f)(x) :=

[∫ΓR0 (x)

∣∣∣t∇e−t2L(f)(y)∣∣∣2 dy dt

t|Q(x, t) ∩ Ω|

] 12

,



where ΓR0(x) is the cone defined by

ΓR0(x) := {(y, t) ∈ Ω× (0, R0] : |y − x| < t}.

When R0 := 1, we denote Sloch,R0

and Sloch,R0

simply, respectively, by Sloch and Sloc

h .To give the definitions of the weighted local Orlicz-Hardy spaces associated with

the Lusin area functions Sloch and Sloc

h , we need the following notation.In what follows, denote by Q the set of all unit cubes of Rn whose interiors are

disjoint. Let Ω be either Rn or a strongly Lipschitz domain of Rn. For any given

Q ∈ Q satisfying Q∩Ω �= ∅, if the center of Q, xQ ∈ Q∩Ω, let Q := Q; if the center

of Q, xQ ∈ Q ∩ Ω�, take the cube Q ⊂ Rn satisfying l(Q) = 2, Q ⊂ Q and the

center of Q, x˜Q ∈ Q∩Ω, where and in what follows, Ω� denotes the complement of

Ω in Rn. Denote by QΩ the set of all cubes Q as above. For any given measurablesubset E of Rn satisfying |E| < ∞ and f ∈ L1

loc(Rn), let

(3.1) mE(f) :=1

|E|

∫E

f(y) dy,

where and in what follows, the space L1loc(R

n) denotes the set of all locally integrablefunctions on Rn.

Definition 3.3. Let Φ satisfy Assumption (A), ω ∈ A∞(Rn), Ω be either Rn or astrongly Lipschitz domain of Rn and L be as in (2.7). A function f ∈ L2(Ω) is said

to be in hΦω, Sh

(Ω) if Sloch (f) ∈ LΦ

ω (Ω) and∑Q∈QΩ

ω (Q ∩ Ω)Φ(mQ∩Ω(|e−L(f)|)

)< ∞.

Furthermore, define

‖f‖hΦω, Sh

(Ω) := inf

{λ ∈ (0,∞) :

∫Ω

Φ

(Sloch (f)(x)

λ

)ω(x) dx ≤ 1

}

+ inf

⎧⎨⎩λ ∈ (0,∞) :∑

Q∈QΩ

ω (Q ∩ Ω)Φ

(mQ∩Ω(|e−L(f)|)

λ

)≤ 1

⎫⎬⎭ .

The weighted local Orlicz-Hardy space hΦSh, ω

(Ω) is defined to be the completion of

hΦSh, ω

(Ω) in the norm ‖ · ‖hΦSh, ω(Ω).

The weighted local Orlicz-Hardy space hΦ˜Sh, ω

(Ω) is defined via replacing Sloch in

the definition of hΦSh, ω

(Ω) by Sloch .

In this section, we present the proof of Theorem 1.4. To this end, we need someauxiliary area functions as follows. Let R0 ∈ (0,∞), α ∈ (0,∞), ε, R ∈ (0, R0) andε < R. For all given f ∈ L2(Ω) and x ∈ Ω, let

Sloc, αh,R0

(f)(x) :=

[∫ΓR0α (x)

∣∣∣t∇e−t2L(f)(y)∣∣∣2 dy dt

t|Q(x, t) ∩ Ω|

] 12

and

Sε,R, αh,R0

(f)(x) :=

[∫Γε, Rα (x)

∣∣∣t∇e−t2L(f)(y)∣∣∣2 dy dt

t|Q(x, t) ∩ Ω|

] 12

,



where and in what follows, for all x ∈ Ω, ΓR0α (x) and Γε, R

α (x) are the local cone andthe truncated cone defined, respectively, by

ΓR0α (x) := {(y, t) ∈ Ω× (0, R0] : |y − x| < αt}

and

Γε, Rα (x) := {(y, t) ∈ Ω× (ε, R] : |y − x| < αt}

for α ∈ (0,∞) and 0 < ε < R < R0. When α = 1, we denote Sloc, αh,R0

(f), Sε,R, αh,R0

(f)

and ΓR0α (x) simply, respectively, by Sloc

h,R0(f), Sε,R

h,R0(f) and ΓR0(x). To show The-

orem 1.4, we first establish the following Proposition 3.4.

Proposition 3.4. Let Φ satisfy Assumption (A), ω ∈ A∞(Rn), R0 ∈ (0,∞) andL be as in (2.7). Let qω, μ and p−Φ be respectively as in (2.5), (2.9) and (2.12). Let

Ω be either Rn or a strongly Lipschitz domain of Rn. Assume that qω, μ and p−Φsatisfy the inequality qω/p

−Φ < (n + μ)/n, and that the semigroup generated by L

has the Gaussian property (G1).(i) If Ω := Rn, then there exists a positive constant C such that for all f ∈

hΦω(R

n) ∩ L2(Rn), ∥∥∥N loc, R0

h (f)∥∥∥LΦ

ω(Rn)≤ C‖f‖hΦ

ω(Rn).

(ii) Under DBC, there exists a positive constant C such that for all f ∈ hΦω, r(Ω)∩

L2(Ω), ∥∥∥N loc, R0

h (f)∥∥∥LΦ

ω(Ω)≤ C‖f‖hΦ

ω, r(Ω).

(iii) Under NBC, there exists a positive constant C such that for all f ∈ hΦω, z(Ω)∩

L2(Ω), ∥∥∥N loc, R0

h (f)∥∥∥LΦ

ω(Ω)≤ C‖f‖hΦ

ω, z(Ω).

To show Proposition 3.4, we need the atomic decomposition characterization ofthe weighted local Orlicz-Hardy space hΦ

ω(Rn) established in [86]. To state this, we

begin with the notions of (ρ, q, s)ω-atoms and the weighted atomic local Orlicz-Hardy space hρ, q, s

ω (Rn).

Definition 3.5. Let Φ satisfy Assumption (A), ω ∈ A∞(Rn), p−Φ , qω and ρ berespectively as in (2.12), (2.5) and (2.14). A triplet (ρ, q, s)ω is called admissibleif q ∈ (qω,∞] and s ∈ Z+ with

s ≥⌊n

(qω

p−Φ− 1

)⌋.

A function a on Rn is called a (ρ, q, s)ω-atom if there exists a cube Q ⊂ Rn suchthat

(i) supp (a) ⊂ Q;(ii)


1q−1[ρ(ω(Q))]−1;

(iii) for all α ∈ Zn+ with |α| ≤ s, when l(Q) < 1,∫

Rn

a(x)xα dx = 0.



Definition 3.6. Let Φ satisfy Assumption (A), ω ∈ A∞(Rn), qω, ρ be respectivelyas in (2.5) and (2.14), and (ρ, q, s)ω admissible. The weighted atomic local Orlicz-Hardy space hρ, q, s

ω (Rn) is defined to be the set of all f ∈ D′(Rn) satisfying that

f =

∞∑i=1

λiai

in D′(Rn), where {ai}i∈N are (ρ, q, s)ω-atoms with supp (ai) ⊂ Qi, {λi}i∈N ⊂ C,and

∞∑i=1

ω(Qi)Φ

(|λi|

ω(Qi)ρ(ω(Qi))

)< ∞.

Moreover, letting

Λ({λiai}i∈N) := inf

{λ ∈ (0,∞) :

∞∑i=1

ω(Qi)Φ

(|λi|

λω(Qi)ρ(ω(Qi))

)≤ 1

},

the quasi-norm of f ∈ hρ, q, sω (Rn) is defined by

‖f‖hρ, q, sω (Rn) := inf {Λ({λiai}i∈N)} ,

where the infimum is taken over all the decompositions of f as above.

The (ρ, q, s)ω-atom and the weighted atomic local Orlicz-Hardy space hρ, q, sω (Rn)

were introduced in [86], in which the following Lemma 3.7 was also obtained (see[86, Theorem 5.6]).

Lemma 3.7. Let Φ satisfy Assumption (A), ω ∈ A∞(Rn), qω, p−Φ and ρ be re-spectively as in (2.5), (2.12) and (2.14). If q ∈ (qω,∞] and integer s satisfiess ≥ �n( qω

p−Φ

− 1)�, then

hρ, q, sω (Rn) = hΦ

ω (Rn)

with equivalent quasi-norms.

We point out that in [86, Theorem 5.6], the assumption p−Φ ∈ (0, 1] was replacedby the assumption pΦ ∈ (0, 1], which, however, can be relaxed; see Remark 2.7.

Now we prove Proposition 3.4 by applying Lemma 3.7.

Proof of Proposition 3.4. We first prove Proposition 3.4(i). To this end, it sufficesto prove that for any (ρ, q, 0)-atom a supported in Q0 := Q(x0, r0) with someq ∈ (qω, ∞] and any λ ∈ C,

(3.2)

∫Rn

Φ(N loc, R0

h (λa)(x))ω(x) dx � ω(Q0)Φ

(|λ|

ω(Q0)ρ(ω(Q0))

).

Indeed, if (3.2) holds, by Lemma 3.7 and an argument in [86, p. 43], we knowthat for any f ∈ hΦ

ω (Rn) ∩ L2(Rn), there exist {λi}i ⊂ C and a sequence {ai}i of

(ρ, q, 0)-atoms such that

(3.3) f =∑i

λiai in L2(Rn)



and

Λ({λiai}i) := inf

{λ ∈ (0,∞) :

∑i

ω(Qi)Φ

(|λi|

λω(Qi)ρ(ω(Qi))

)≤ 1

}(3.4)

�‖f‖hΦω(Rn),

where for each i, supp (ai) ⊂ Qi. By (3.3), we conclude that for all x ∈ Rn,

N loc, R0

h (f)(x) ≤∑i

N loc, R0

h (λiai)(x),

which, together with the assumptions that Φ is strictly increasing, subadditive andcontinuous, and that for any λ ∈ (0,∞) and each i,

N loc, R0

h (f/λ) = N loc, R0

h (f)/λ, N loc, R0

h (ai/λ) = N loc, R0

h (ai)/λ

and (3.2), implies that for any λ ∈ (0,∞),∫Rn

Φ

(N loc, R0

h (f)(x)

λ

)ω(x) dx

=

∫Rn

Φ

(N loc, R0

h

(f

λ

)(x)

)ω(x) dx

≤∑i

∫Rn

Φ

(N loc, R0

h

(λiaiλ

)(x)

)ω(x) dx

�∑i

ω(Qi)Φ

(|λi|

λω(Qi)ρ(ω(Qi))

).

From this and (3.4), we deduce that∥∥∥N loc, R0

h (f)∥∥∥LΦ

ω(Rn)� ‖f‖hΦ

ω(Rn),

which is as desired.Now we prove (3.2). Let Q0 := 4(R0 + 1)Q0. Then we write∫

Rn

Φ(N loc, R0

h (λa)(x))ω(x) dx(3.5)

=

∫˜Q0

Φ(N loc, R0

h (λa)(x))ω(x) dx+

∫˜Q�0

· · · =: I1 + I2.

We first estimate I1. From the assumption that qωp−Φ

< n+μn , we deduce that there

exist q ∈ (qω,∞), p0 ∈ (0, p−Φ) and μ ∈ (0, μ) such that Φ is of lower type p0 and

q

p0<

n+ μ

n.(3.6)

Since q ∈ (qω,∞), by the definition of qω, we know that ω ∈ Aq(Rn). Since Φ

is concave, by Jensen’s inequality, Holder’s inequality, (vii) and (iii) of Lemma 2.2,we have

I1 ≤ ω(Q0)Φ

(1

ω(Q0)

∫˜Q0

N loc, R0

h (λa)(x)ω(x) dx

)(3.7)

≤ ω(Q0)Φ

(1

[ω(Q0)]1q

{∫˜Q0

[N loc, R0

h (λa)(x)]q

ω(x) dx

} 1q

)



� ω(Q0)Φ

(1

[ω(Q0)]1q

|λ|‖a‖Lqω(Rn)

)

� ω(Q0)Φ

(|λ|

ω(Q0)ρ(ω(Q0))

)� ω(Q0)Φ

(|λ|

ω(Q0)ρ(ω(Q0))

),

which is as desired.Now we deal with I2 by considering the following two cases for Q0.

Case 1) l(Q0) < 1. In this case, let x ∈ (Q0)�, t ∈ (0, R0] and y ∈ Rn satisfy

|x− y| < t. Let μ be as in (3.6). By an argument similar to that in [87, p. 18], weknow that for all z ∈ Q0,

|Kt2(y, z)−Kt2(y, x0)| �|z − x0|μ

|x− x0|n+μ,

which, together with the moment condition of a, Holder’s inequality and (2.1),implies that∣∣∣e−t2L(λa)(y)

∣∣∣ = ∣∣∣∣∫Rn

λ [Kt2(y, z)−Kt2(y, x0)] a(z) dz

∣∣∣∣(3.8)

�∫Rn

|λ||z − x0|μ|x− x0|n+μ

|a(z)| dz

� |λ||Q0|μ/n‖a‖Lqω(Rn)

{∫Q0

[ω(x)]−q′/q dx

} 1q′

|x− x0|−(n+μ)

� |λ||Q0|μn ‖a‖Lq

ω(Rn)

|Q0|[ω(Q0)]

1q

|x− x0|−(n+μ)

� |λ||Q0|n+μn

ω(Q0)ρ(ω(Q0))|x− x0|−(n+μ),

where and in what follows, 1q + 1

q′ = 1. From this, it further follows that for all

x ∈ (Q0)�,

N loc, R0

h (λa)(x) � |λ||Q0|n+μn

ω(Q0)ρ(ω(Q0))|x− x0|−(n+μ),

which, together with the lower type p0 and upper type 1 properties of Φ, Lemma2.2(iii) and (3.6), implies that

I2 �∫( ˜Q0)�

Φ

(|λ||Q0|(n+μ)/n

ω(Q0)ρ(ω(Q0))|x− x0|−(n+μ)

)ω(x) dx(3.9)

�∞∑k=2

∫2k+1Q0\2kQ0

Φ(2−k(n+μ)|λ|[ω(Q0)ρ(ω(Q0))]

−1)ω(x) dx

�∞∑k=2

2−k[(n+μ)p0−nq]ω(Q0)Φ

(|λ|

ω(Q0)ρ(ω(Q0))

)� ω(Q0)Φ

(|λ|

ω(Q0)ρ(ω(Q0))

).



Case 2) l(Q0) ≥ 1. In this case, let x ∈ (Q0)�, t ∈ (0, R0] and y ∈ Rn satisfy

|x− y| < t. Then for all z ∈ Q0,

|y − z| ≥ |x− z| − |x− y| ≥ |x− x0| − |x0 − z| − |x− y|

≥ |x− x0| −1

4(R0 + 1)|x− x0| −

R0

2(R0 + 1)|x− x0| >

1

2|x− x0|,

which, together with (2.8), Holder’s inequality and (2.1), implies that∣∣∣e−t2L(λa)(y)∣∣∣ = ∣∣∣∣∫

Q0

λKt2(y, z)a(z) dz

∣∣∣∣ � ∫Q0

|λ|t|y − z|n+1

|a(z)| dz

� |λ||x− x0|n+1

∫Q0

|a(z)| dz � |λ|‖a‖Lqω(Rn)

|Q0|[ω(Q0)]

1q

1

|x− x0|n+1

� |λ||Q0|[ω(Q0)ρ(ω(Q0))]−1|x− x0|−(n+1),

where we omitted some computations similar to (3.8). From this, we infer that for

all x ∈ (Q0)�,

N loc, R0

h (λa)(x) � |λ||Q0|[ω(Q0)ρ(ω(Q0))]−1|x− x0|−(n+1),

which, together with the lower type p0 and upper type 1 properties of Φ, Lemma2.2(iii), l(Q0) ≥ 1 and (3.6), implies that

I2 �∫( ˜Q0)�

Φ(|λ||Q0|[ω(Q0)ρ(ω(Q0))]

−1|x− x0|−(n+1))ω(x) dx(3.10)

�∞∑k=2

∫2k+1Q0\2kQ0

Φ(2−k(n+1)|λ|[ω(Q0)ρ(ω(Q0))]

−1)ω(x) dx

�∞∑k=2

2−k[(n+1)p0−nq]ω(Q0)Φ

(|λ|

ω(Q0)ρ(ω(Q0))

)� ω(Q0)Φ

(|λ|

ω(Q0)ρ(ω(Q0))

).

Thus, by (3.5), (3.7), (3.9) and (3.10), we conclude that∫Rn

Φ(N loc, R0


(|λ|

ω(Q0)ρ(ω(Q0))

),

which implies that (3.2) holds and hence completes the proof of Proposition 3.4(i).Now we prove Proposition 3.4(ii). Let f ∈ hΦ

ω, r(Ω)∩L2(Ω). By the definition of

hΦω, r(Ω), we know that there exists f ∈ hΦ

ω (Rn) such that f

∣∣Ω= f and

(3.11)∥∥∥f∥∥∥

hΦω(Rn)

� ‖f‖hΦω, r(Ω).

Let q be as in (3.6). To show Proposition 3.4(ii), we only need to prove that forany (ρ, q, 0)ω-atom a supported in Q0 := Q(x0, r0) and λ ∈ C,

(3.12)

∫Ω

Φ(N loc, R0


(|λ|

ω(Q0)ρ(ω(Q0))

).

Indeed, since f ∈ hΦω (R

n), by Lemma 3.7, there exist {λi}i ⊂ C and a sequence

{ai}i of (ρ, q, 0)-atoms such that f =∑

i λiai in D′(Rn) and

Λ ({λiai}i) ∼∥∥∥f∥∥∥

hΦω(Rn)

.



Moreover, by the proof of [86, Theorem 5.1], we know that the supports of {ai}iare of finite intersection property. By this, f ∈ L2(Ω),

f =∑i

λiai

in D′(Rn) and f∣∣Ω= f , we see that f =

∑i λiai almost everywhere on Ω, which

further implies that∫Ω

Kt2(x, y)f(y) dy =∑i

λi

∫Ω

Kt2(x, y)ai(y) dy.

From this, we deduce that for all x ∈ Ω,

N loc, R0

h (f)(x) ≤∑i

N loc, R0

h (λiai)(x),

which, together with the assumptions that Φ is strictly increasing, subadditive andcontinuous, and that for any λ ∈ (0,∞) and each i,

N loc, R0

h (f/λ) = N loc, R0

h (f)/λ, N loc, R0

h (ai/λ) = N loc, R0

h (ai)/λ

and (3.12), implies that for any λ ∈ (0,∞),∫Ω

Φ

(N loc, R0

h (f)(x)

λ

)ω(x) dx

=

∫Ω

Φ

(N loc, R0

h

(f

λ

)(x)

)ω(x) dx

≤∑i

∫Ω

Φ

(N loc, R0

h

(λiaiλ

)(x)

)ω(x) dx

�∑i

ω(Qi)Φ

(|λi|

λω(Qi)ρ(ω(Qi))

),

where for each i, supp (ai) ⊂ Qi. From this, Lemma 3.7 and (3.11), we deduce that∥∥∥N loc, R0

h (f)∥∥∥LΦ

ω(Ω)� Λ({λiai}i) ∼

∥∥∥f∥∥∥hΦω(Rn)

� ‖f‖hΦω, r(Ω),

which, together with the arbitrariness of f ∈ hΦω, r(Ω) ∩ L2(Ω), implies the conclu-

sions of Proposition 3.4(ii).It is easy to see that for all x ∈ Ω,

(3.13) e−t2L(λa)(x) =

∫Q0∩Ω

λKt2(x, y)a(y) dy.

Now we show (3.12) by considering the following three cases for Q0.

Case i) Q0 ∩ Ω = ∅. In this case, by (3.13), we know that N loc, R0

h (λa)(x) = 0 forall x ∈ Ω. From this, it follows that (3.12) holds.

Case ii) Q0 ⊂ Ω. In this case, the proof of (3.12) is similar to that of (3.2). Weomit the details.

Case iii) Q0 ∩ ∂Ω �= ∅. In this case, recall that for any x ∈ Ω, t ∈ (0,∞) andy ∈ ∂Ω, Kt(x, y) = 0 (see, for example, [8, p. 156]). Take y0 ∈ Q0 ∩ ∂Ω. Then we



see that for any x ∈ Ω and t ∈ (0, R0], Kt2(x, y0) = 0, which further implies thatfor any x ∈ Ω,

e−t2L(λa)(x) =

∫Q0∩Ω

λ [Kt2(x, y)−Kt2(x, y0)] a(y) dy,

provided l(Q0) < 1. The remaining estimates are similar to those of Proposition3.4(i). We omit the details, which completes the proof of Proposition 3.4(ii).

Finally we give the proof of Proposition 3.4(iii). Let f ∈ hΦω, z(Ω) ∩ L2(Ω) and

f be the zero extension out of Ω of f . Then f ∈ hΦω, z(R

n) ∩ L2(Rn). For allt ∈ (0, R0], x ∈ Ω and y ∈ Ω, define

Fx, t(y) := tn(1 +

|x− y|t

)n+1

Kt2(x, y).

It was proved in [8, p. 156] that Fx, t can be extended to a bounded Holder contin-uous function on Rn with the Holder index μ ∈ (0, μ). We denote this extension by

Fx, t. For all t ∈ (0, R0], x ∈ Ω and y ∈ Rn, define

Kt2(x, y) := tn(1 +

|x− y|t

)−(n+1)

Fx, t(y).

It was also proved in [8, p. 157] that Kt2 satisfies (2.8) and (2.9) with μ replacedby any μ ∈ (0, μ). Obviously, for all t ∈ (0,∞) and all x, y ∈ Ω,

Kt2(x, y) = Kt2(x, y).

By Lemma 3.7 and an argument in [86, p. 43], we know that there exist {λi}i ⊂ C

and a sequence {ai}i of (ρ, q, 0)ω-atoms, with q ∈ (qω,∞], such that

f =∑i

λiai

in L2(Rn). Thus, for all t ∈ (0, R0] and x ∈ Ω,

e−t2L(f)(x) =

∫Ω

Kt2(x, y)f(y) dy

=

∫Rn

Kt2(x, y)f(y) dy =∑i

∫Rn

λiKt2(x, y)ai(y) dy,

which implies that

N loc, R0

h (f)(x) ≤∑i

N loc, R0

h (λiai)(x).

The remainder of the proof is similar to that of Proposition 3.4(i). We omit thedetails. This finishes the proof of Proposition 3.4. �

To show Theorem 1.4, we need the following key proposition.

Proposition 3.8. Let Φ, Ω, ω and L be as in Proposition 3.4, and R0 ∈ (0,∞).Then there exists a positive constant C such that for all f ∈ L2(Ω) satisfying∥∥∥N loc, 2R0

h (f)∥∥∥LΦ

ω(Ω)< ∞,∥∥∥Sloc

h,R0(f)

∥∥∥LΦ

ω(Ω)≤ C

∥∥∥N loc, 2R0

h (f)∥∥∥LΦ

ω(Ω).



To show Proposition 3.8, we need the following Lemmas 3.9 through 3.11.In [8, p. 183], Auscher and Russ proved the following geometric property of

strongly Lipschitz domains, which plays an important role in this paper.

Lemma 3.9. Let Ω be a strongly Lipschitz domain of Rn. Then there exists aconstant C ∈ (0, 1] such that for all cubes Q centered in Ω with

l(Q) ∈ (0, ∞) ∩ (0, 2 diam (Ω)],

|Q ∩ Ω| ≥ C|Q|.

In what follows, we denote by B((z, τ ), r) the ball in Rn × (0,∞) with center(z, τ ) and radius r; namely,

B((z, τ ), r) := {(x, t) ∈ Rn × (0,∞) : max(|x− z|, |t− τ |) < r}.

Lemma 3.10. Let Φ satisfy Assumption (A), ω ∈ A∞(Rn), R0 ∈ (0,∞) and Lbe as in (2.7). Let Ω be either Rn or a strongly Lipschitz domain of Rn. Assumethat the semigroup generated by L has the Gaussian property (G1). Then thereexist ε0 ∈ (0,∞) and a positive constant C such that for all γ ∈ (0, 1], λ ∈ (0,∞),

ε, R ∈ (0, R0) with ε < R and f ∈ L2(Ω) satisfying ‖N loc, 2R0

h (f)‖LΦω(Ω) < ∞,

ω({

x ∈ Ω : Sε,R, 1

20

h,R0(f)(x) > 2λ, N loc, 2R0

h (f)(x) ≤ γλ})

(3.14)

≤ Cγε0ω({

x ∈ Ω : Sε, R, 1

2

h,R0(f)(x) > λ

}).

We point out that in the proof of Proposition 3.8, Lemma 3.10 plays a keyrole. The inequality (3.14) is usually called the “good-λ inequality” concerning the

maximal function N loc, 2R0

h (f) and the truncated area functions Sε, R, 1

20

h,R0(f) and

Sε,R, 1

2

h,R0(f).

Proof of Lemma 3.10. To prove this lemma, we borrow some ideas from [7] and [8].Fix 0 < ε < R < R0, γ ∈ (0, 1] and λ ∈ (0,∞). Let f ∈ L2(Ω) satisfy∥∥∥N loc, 2R0

h (f)∥∥∥LΦ

ω(Ω)< ∞

and

O :={x ∈ Ω : S

ε, R, 12

h,R0(f)(x) > λ

}.

It is easy to show that O is an open subset of Ω.Now we show (3.14) by considering the following two cases for O.

Case 1) O �= Ω. In this case, let

O =⋃k

(Qk ∩ Ω)(3.15)

be the Whitney decomposition of O, where {Qk}k are dyadic cubes of Rn withdisjoint interiors and 2Qk ∩ Ω ⊂ O ⊂ Ω, but

4Qk ∩ Ω ∩ (Ω \O) �= ∅.



To show (3.14), by (3.15) and the disjoint property of {Qk}k, it suffices to showthat for all k,

ω({

x ∈ Qk ∩ Ω : Sε, R, 1

20

h,R0(f)(x) > 2λ, N loc, 2R0

h (f)(x) ≤ γλ})

(3.16)

� γε0ω(Qk ∩ Ω).

From now on, we fix k and denote by lk the sidelength of Qk.If x ∈ Qk ∩ Ω, then

Smax{10lk, ε}, R, 1

20

h,R0(f)(x) ≤ λ.(3.17)

Indeed, pick xk ∈ 4Qk ∩Ω with xk �∈ O. For any (y, t) ∈ Ω× (0, R0], if |x− y| < t20

and t ≥ max{10lk, ε}, then

|xk − y| ≤ |xk − x|+ |x− y| < 4lk +t

20<

t

2,

which implies that

Γmax{10lk, ε}, R120

(x) ⊂ Γmax{10lk, ε}, R12

(xk).

By this, we obtain that

Smax{10lk, ε}, R, 1

20

h,R0(f)(x) ≤ S

max{10lk, ε}, R, 12

h,R0(f)(xk) ≤ λ.

Thus, the claim (3.17) holds.If ε ≥ 10lk, by (3.17), we see that (3.16) trivially holds. If ε < 10lk, to show

(3.16), by the fact that

Sε,R, 1

20

h,R0(f) ≤ S

ε, 10lk,120

h,R0(f) + S

10lk, R, 120

h,R0(f)

and (3.17), it remains to show that

ω({x ∈ Qk ∩ F : g(x) > λ}) � γε0ω(Qk ∩ Ω),(3.18)

where g := Sε, 10lk,

120

h,R0(f) and

F := {x ∈ Ω : N loc, 2R0

h (f)(x) ≤ γλ}.Now we prove (3.18). Similar to the proof of [87, (3.18)], we have

|{x ∈ Ω ∩Qk : g(x) > λ}| � γ2|Qk ∩ Ω|.(3.19)

By ω ∈ A∞(Rn), Lemma 2.2(vi), we know that there exists r ∈ (1,∞] such thatω ∈ RHr(R

n), which, together with Lemma 2.2(v), Lemma 3.9 and (3.19), impliesthat

ω({x ∈ Qk ∩ F : g(x) > λ})ω(Qk)

�(|{x ∈ Qk ∩ F : g(x) > λ}|

|Qk|

) r−1r

(3.20)

�(γ2|Qk ∩ Ω|

|Qk|

) r−1r

∼ γ2(r−1)

r .

Take ε0 := 2(r − 1)/r. Then from (3.20) and Lemmas 3.9 and 2.2(iii), we deducethat

ω({x ∈ Qk ∩ F : g(x) > λ}) � γε0ω(Qk) ∼ γε0ω(Qk ∩ Ω),

which implies that (3.18) holds. Thus, (3.14) holds in the case that O �= Ω.



Case 2) O = Ω. In this case, similar to the proof of [87, Lemma 3.5], we know thatΩ is bounded.

Also, similar to the proof of [87, Lemma 3.5], we know that there exists a positiveconstant C1 such that for all R ∈ (R0/2, R0) and x ∈ Ω,

(3.21) SR0/2, R, 1

20

h,R0(f)(x) ≤ C1N loc, 2R0

h (f)(x).

Now we continue the proof of Lemma 3.10 by using (3.21). Without loss ofgenerality, we may assume that R ≥ R0/2. Otherwise, we replace R just by R0/2in (3.15). If γ ≥ 1

C1, then

ω({

x ∈ Ω : Sε,R, 1

20

h,R0(f)(x) > 2λ, N loc, 2R0

h (f)(x) ≤ γλ})

≤ ω(Ω) ≤ Cε01 γε0ω(O) � γε0ω(O),

which shows (3.14) in the case that O = Ω and γ ≥ 1C1

.

If γ < 1C1

, by the fact that N loc, 2R0

h (f)(x) ≤ γλ for all x ∈ F and (3.21), we

conclude that for any R ≥ R0/2 and x ∈ F ,

SR0/2, R, 1

20

h,R0(f)(x) ≤ C1N loc, 2R0

h (f)(x) <1

γγλ = λ,

which implies that{x ∈ Ω : S

ε, R, 120

h,R0(f)(x) > 2λ, N loc, 2R0

h (f)(x) ≤ γλ}

(3.22)

⊂{x ∈ Ω : S

ε, R0/2,120

h,R0(f)(x) > λ, N loc, 2R0

h (f)(x) ≤ γλ}.

Thus, to finish the proof of (3.14) in this case, it suffices to show that

(3.23)∣∣∣{x ∈ Ω : S

ε, R0/2,120

h,R0(f)(x) > λ, N loc, 2R0

h (f)(x) ≤ γλ}∣∣∣ � γ2|O|.

Indeed, if (3.23) holds, by Lemma 3.9 and the assumption that Ω is bounded, weknow that there exists a cube Q0 ⊂ Rn such that Ω ⊂ Q0 and |Q0| ∼ |Ω|. Fromthis, O = Ω, (3.22), (3.23), and (iii) and (v) of Lemma 2.2, we infer that

ω({x ∈ Ω : Sε, R, 1

20

h,R0(f)(x) > 2λ, N loc, 2R0

h (f)(x) ≤ γλ})ω(Q0)

≤ω({x ∈ Ω : S

ε, R0/2,120

h,R0(f)(x) > λ, N loc, 2R0

h (f)(x) ≤ γλ})ω(Q0)

�

⎧⎨⎩ |{x ∈ Ω : Sε, R0/2,

120

h,R0(f)(x) > λ, N loc, 2R0

h (f)(x) ≤ γλ}||Q0|

⎫⎬⎭r−1r

�{γ2|Ω||Q0|

} r−1r

∼ γε0 .

By this, the fact that |Q0| ∼ |Ω|, (iii) and (v) of Lemma 2.2 and O = Ω, we concludethat

ω({

x ∈ Ω : Sε, R, 1

20

h,R0(f)(x) > 2λ, N loc, 2R0

h (f)(x) ≤ γλ})

� γε0ω(Q0) ∼ γε0ω(O),

which shows (3.14) in the case that O = Ω and γ < 1C1

.



Finally, we point out that the proof of (3.23) is similar to that of (3.19) with10lk and Qk ∩F , respectively, replaced by R0/2 and Ω. We omit the details, whichcompletes the proof of Lemma 3.10. �

Lemma 3.11. Let Φ, Ω and L be as in Proposition 3.4. Let ω ∈ A∞(Rn) andR0 ∈ (0,∞). For all α, β ∈ (0,∞), 0 < ε < R < R0 and all f ∈ L2(Ω),∫

Ω

Φ(Sε, R, αh,R0

(f)(x))ω(x) dx ∼

∫Ω

Φ(Sε, R, βh,R0

(f)(x))ω(x) dx,

where the implicit constants are independent of ε, R and f .

The proof of Lemma 3.11 is similar to that of [20, Proposition 4]. We omit thedetails.

Now we show Proposition 3.8 by virtue of Lemmas 3.10 and 3.11.

Proof of Proposition 3.8. Let f ∈ L2(Ω) satisfy ‖N loc, 2R0

h (f)‖LΦω(Ω) < ∞. By the

upper type 1 and the lower type p0 properties of Φ, where p0 ∈ (0, p−Φ), we knowthat for all t ∈ (0,∞),

Φ(t) ∼∫ t

0

Φ(s)

sds.

From this, Fubini’s theorem and Lemma 3.10, it follows that for all ε, R ∈ (0, R0]with ε < R and γ ∈ (0, 1],∫

Ω

Φ(Sε, R, 1

20

h,R0(f)(x)

)ω(x) dx(3.24)

∼∫Ω

⎧⎨⎩∫

˜Sε, R, 1

20h,R0

(f)(x)

0

Φ(t)

tdt

⎫⎬⎭ω(x) dx ∼∫ ∞

0

Φ(t)

tσ˜Sε, R, 1

20h,R0

(f)(t) dt

�∫ ∞

0

Φ(t)

tσN loc, 2R0

h (f)(γt) dt+ γε0

∫ ∞

0

Φ(t)

tσ˜Sε, R, 1

2h,R0

(f)(t/2) dt

� 1

γ

∫ ∞

0

Φ(t)

tσN loc, 2R0

h (f)(t) dt+ γε0

∫ ∞

0

Φ(t)

tσ˜Sε, R, 1

2h,R0

(f)(t) dt

∼ 1

γ

∫Ω

Φ(N loc, 2R0

h (f)(x))ω(x) dx+ γε0

∫Ω

Φ(Sε, R, 1

2

h,R0(f)(x)

)ω(x) dx,

where

σ˜Sε, R, 1

20h,R0

(f)(t) := ω

({x ∈ Ω : S

ε, R, 120

h,R0(f)(x) > t

})and ε0 is as in Lemma 3.10.

Furthermore, by Lemma 3.11, (3.24) and Sε, R, 1

2

h,R0(f) ≤ Sε, R

h,R0(f), we know that

for all ε, R ∈ (0, R0] with ε < R and γ ∈ (0, 1],∫Ω

Φ(Sε, Rh,R0

(f)(x))ω(x) dx ∼

∫Ω

Φ(Sε, R, 1

20

h,R0(f)(x)

)ω(x) dx

� 1

γ

∫Ω

Φ(N loc, 2R0

h (f)(x))ω(x) dx

+ γε0

∫Ω

Φ(Sε, Rh,R0

(f)(x))ω(x) dx,



which, together with the facts that for all λ ∈ (0,∞),

Sε, Rh,R0

(f/λ) = Sε, Rh,R0

(f)/λ and N loc, 2R0

h (f/λ) = N loc, 2R0

h (f)/λ,

implies that there exists a positive constant C2 such that∫Ω

Φ

(Sε, Rh,R0

(f)(x)

λ

)ω(x) dx(3.25)

≤ C2

{1

γ

∫Ω

Φ

(N loc, 2R0

h (f)(x)

λ

)ω(x) dx

+γε0

∫Ω

Φ

(Sε,Rh,R0

(f)(x)

λ

)ω(x) dx

}.

Take γ ∈ (0, 1] such that C2γε0 = 1

2 . Then from (3.25), it follows that for allλ ∈ (0,∞),∫

Ω

Φ

(Sε, Rh,R0

(f)(x)

λ

)ω(x) dx �

∫Ω

Φ

(N loc, 2R0

h (f)(x)

λ

)ω(x) dx.

By the Fatou lemma and letting ε → 0 and R → R0, we see that for any λ ∈ (0,∞),∫Ω

Φ

(Sloch,R0

(f)(x)

λ

)ω(x) dx �

∫Ω

Φ

(N loc, 2R0

h (f)(x)

λ

)ω(x) dx,

which implies that ∥∥∥Sloch,R0

(f)∥∥∥LΦ

ω(Ω)�∥∥∥N loc, 2R0

h (f)∥∥∥LΦ

ω(Ω).

This finishes the proof of Proposition 3.8. �

Proposition 3.12. Let Φ, Ω, ω and L be as in Proposition 3.4 and R0 ∈ (0,∞).Then there exists a positive constant C such that for all f ∈ L2(Ω),∥∥Sloc

h,R0(f)

∥∥LΦ

ω(Ω)≤ C

∥∥∥Sloch,R0

(f)∥∥∥LΦ

ω(Ω).

Proof. Let f ∈ L2(Ω), x ∈ Ω and p0 ∈ (0, p−Φ). Then by the definition of p−Φ ,we know that Φ is of lower type p0. Similar to the proof of [87, (3.38)], we seethat there exists a positive constant C3, independent of f and x, such that for allε ∈ (0, 1),

Sloch,R0

(f)(x) ≤ C3

εSloc, 3

2

h,R0(f)(x) + εSloc, 2

h,R0(f)(x).(3.26)

Also, similar to the proof of Lemma 3.11, we conclude that there exists a positiveconstant C4 such that for all g ∈ L2(Ω),∫

Ω

Φ(Sloc, 2h,R0

(g)(y))ω(y) dy ≤ C4

∫Ω

Φ(Sloch,R0

(g)(y))ω(y) dy.

From this, (3.26), the lower type p0 and the upper type 1 properties of Φ, it follows

that there exists a positive constant C such that∫Ω

Φ(Sloch,R0

(f)(x))ω(x) dx(3.27)



≤∫Ω

Φ

(C3

εSloc, 3

2

h,R0(f)(x)

)ω(x) dx+

∫Ω

Φ(εSloc, 2

h,R0(f)(x)

)ω(x) dx

≤ C

{C3

ε

∫Ω

Φ(S

32

h,R0(f)(x)

)ω(x) dx

+C4εp0

∫Ω

Φ(Sloch,R0

(f)(x))ω(x) dx

}.

Take ε ∈ (0, 1) small enough such that CC4εp0 ≤ 1

2 . By this, (3.27) and Lemma3.11, we see that∫

Ω

Φ(Sloch,R0

(f)(x))ω(x) dx �

∫Ω

Φ(Sloch,R0

(f)(x))ω(x) dx,

which, together with the facts that for all λ ∈ (0,∞),

Sloch,R0

(f/λ) = Sloch,R0

(f)/λ and Sloch,R0

(f/λ) = Sloch,R0

(f)/λ,

implies that∫Ω

Φ

(Sloch,R0

(f)(x)

λ

)ω(x) dx �

∫Ω

Φ

(Sloch,R0

(f)(x)

λ

)ω(x) dx.

From this, the desired conclusion follows, which completes the proof of Proposi-tion 3.12. �

To complete the proof of Theorem 1.4, we need the following key proposition.

Proposition 3.13. Let Φ, Ω, L, ω, p−Φ , p+Φ and rω be as in Theorem 1.4, andR0 ∈ (0,∞). Assume that the semigroup generated by L has the Gaussian property(G1).

(i) If Ω := Rn, then there exists a positive constant C such that for all f ∈ L2(Rn)satisfying ‖Sloc

h,R0(f)‖LΦ

ω(Rn) < ∞,

‖f‖hΦω(Rn) ≤ C

⎡⎣ ∥∥Sloch,R0

(f)∥∥LΦ

ω(Rn)

+ inf

⎧⎨⎩λ ∈ (0,∞) :∑

Qk∈Qω(Qk)Φ

(mQk

(|e−R20L(f)|)

λ

)≤ 1

⎫⎬⎭⎤⎦ .

(ii) Under DBC, if Ω� is unbounded, then there exists a positive constant C suchthat for all f ∈ L2(Ω) satisfying ‖Sloc

h,R0(f)‖LΦ

ω(Ω) < ∞,

‖f‖hΦω, r(Ω) ≤ C

⎡⎣∥∥Sloch,R0

(f)∥∥LΦ

ω(Ω)+ inf

⎧⎨⎩λ ∈ (0,∞) :

∑˜Qk∈QΩ

ω(Qk ∩ Ω

)Φ

(m

˜Qk∩Ω(|e−R20L(f)|)

λ

)≤ 1

⎫⎬⎭⎤⎦ .



(iii) Under NBC, there exists a positive constant C such that for all f ∈ L2(Ω)satisfying ‖Sloc

h,R0(f)‖LΦ

ω(Ω) < ∞,

‖f‖hΦω, r(Ω) ≤ C

⎡⎣∥∥Sloch,R0

(f)∥∥LΦ

ω(Ω)+ inf

⎧⎨⎩λ ∈ (0,∞) :

∑˜Qk∈QΩ

ω(Qk ∩ Ω

)Φ

(m

˜Qk∩Ω(|e−R20L(f)|)

λ

)≤ 1

⎫⎬⎭⎤⎦ .

Let Ω be either Rn or a strongly Lipschitz domain of Rn. To show Proposition3.13, we need the atomic decomposition of functions in the local tent space onΩ. Now we recall some definitions and notation about the tent space, which wasinitially introduced by Coifman, Meyer and Stein [20] on Rn, and then generalizedby Russ [73] to spaces of homogeneous type in the sense of Coifman and Weiss[21, 22]. Recently, Carbonaro, McIntosh and Morris [14] developed the theory ofthe local tent space on locally doubling metric measure spaces. Recall that it iswell known that the strongly Lipschitz domain Ω is a space of homogeneous type.

Let R0 ∈ (0,∞) and ω ∈ A∞(Rn). For all measurable functions g on Ω× (0, R0]and x ∈ Ω, define

Aloc, R0(g)(x) :=

[∫ΓR0(x)

|g(x, t)|2 dy

|Q(x, t) ∩ Ω|dt

t

] 12

,

where

ΓR0(x) := {(y, t) ∈ Ω× (0, R0] : |y − x| < t}.In what follows, we denote by T loc, R0

Φ, ω (Ω) the space of all measurable functions g

on Ω× (0, R0] such that Aloc, R0(g) ∈ LΦω (Ω) and for any g ∈ T loc, R0

Φ, ω (Ω), define itsnorm by

‖g‖T

loc, R0Φ, ω (Ω)

:=∥∥Aloc, R0(g)

∥∥LΦ

ω(Ω)

:= inf

{λ ∈ (0,∞) :

∫Ω

Φ

(Aloc, R0(g)(x)

λ

)ω(x) dx ≤ 1

}.

A function a on Ω× (0, R0] is called a T loc, R0

Φ, ω (Ω)-atom if

(i) there exists a cube Q := Q(xQ, l(Q)) ⊂ Rn with xQ ∈ Ω and

l(Q) ∈ (0,∞) ∩ (0, diam (Ω)]

such that supp (a) ⊂ Q ∩ Ω, where and in what follows,

Q ∩ Ω :=

{(y, t) ∈ Ω× (0, R0] : |y − xQ| <

l(Q)

2− t

};

(ii)

‖a‖2T 22 (Ω×(0,R0])

:=

∫Q∩Ω

|a(y, t)|2 dy dtt

≤ |Q ∩ Ω|[ω(Q ∩ Ω)ρ(ω(Q ∩ Ω))]−2.

Since Φ is concave, by Jensen’s inequality, it is easy to see that for all T loc, R0

Φ, ω (Ω)-atoms a, we have

‖a‖T

loc, R0Φ, ω (Ω)

≤ 1;



see [49]. By a slight modification on the proof of [48, Theorem 3.1], we have the

following atomic decomposition for functions in T loc, R0

Φ, ω (Ω). We omit the details.

Lemma 3.14. Let Φ satisfy Assumption (A), Ω be either Rn or a strongly Lipschitz

domain of Rn and ω ∈ A∞(Rn). Then for all f ∈ T loc, R0

Φ, ω (Ω), there exist a sequence

{aj}j of T loc, R0

Φ, ω (Ω)-atoms and a sequence {λj}j ⊂ C such that for almost every

(x, t) ∈ Ω× (0, R0],

f(x, t) =∑j

λjaj(x, t).

Moreover, there exists a positive constant C such that for all f ∈ T loc, R0

Φ, ω (Ω),

Λ({λjaj}j)

:= inf

⎧⎨⎩λ ∈ (0,∞) :∑j

ω(Qj ∩ Ω)Φ

(|λj |

λω(Qj ∩ Ω)ρ(ω(Qj ∩ Ω))

)≤ 1

⎫⎬⎭≤ C‖f‖

Tloc, R0Φ, ω (Ω)

,

where for each j, Qj ∩ Ω appears in the support of aj.

In [8, p. 183], Auscher and Russ showed the following property of strongly Lips-chitz domains, which plays an important role in the proof of Proposition 3.13.

Lemma 3.15. Let Ω be a strongly Lipschitz domain of Rn. Then there existsγΩ ∈ (0,∞) such that for any cube Q satisfying l(Q) < γΩ and 2Q ⊂ Ω but

4Q∩∂Ω �= ∅, where ∂Ω denotes the boundary of Ω, there exists a cube Q ⊂ Ω� such

that l(Q) = l(Q) and the distance from Q to Q is comparable to l(Q). Furthermore,

γΩ = ∞ if Ω� is unbounded.

Now we show Proposition 3.13 by applying Lemmas 3.7, 3.14 and 3.15.

Proof of Proposition 3.13. We first prove Proposition 3.13(i) by borrowing someideas from the proof of [22, p. 594, Theorem C] (see also [42] and [51]). Let a be a

T loc, R0

Φ, ω (Rn)-atom, supp (a) ⊂ Q with Q := Q(x0, r0), and

(3.28) α := 8

∫ R0

0

t2Le−t2L(a)dt

t.

Set Rk(Q) := 2k+1Q \ 2kQ when k ∈ N and R0(Q) := 2Q. For k ∈ Z+, letχk := χRk(Q), χk := |Rk(Q)|−1χk,

mk :=

∫Rk(Q)

α(x) dx

and Mk := αχk −mkχk. Then we have

(3.29) α =

∞∑k=0

Mk +

∞∑k=0

mkχk.

For j ∈ Z+, let

Nj :=

∞∑k=j

mk.



By [8, Lemma A.5(a)], we see that∫Rn

α(x) dx = 0,

which, together with (3.29), yields that

(3.30) α =∞∑k=0

Mk +∞∑k=0

Nk+1 (χk+1 − χk) .

Obviously, for all k ∈ Z+,

(3.31)

∫Rn

Mk(x) dx = 0.

In what follows, if (2.10) holds with p+Φ for all t ∈ [1,∞) and s ∈ (0,∞), then we

choose pΦ := p+Φ ; otherwise, since Φ is concave, we know p+Φ < 1 and we choose

pΦ ∈ (p+Φ , 1) to be close enough to p+Φ . Then we know that Φ has the upper typepΦ property. From the hypotheses

qω

p−Φ<

n+ μ

n,2qω

p−Φ<

n+ 1

n+

rω − 1

p+Φrωand rω >

2

2− qω,

we deduce that there exist p0 ∈ (0, p−Φ), r0 ∈ (1, rω), and q1, q2, q3 ∈ (qω, 2) suchthat

q1p0

<n+ μ

n,2q2p0

<n+ 1

n+

r0 − 1

pΦr0and

2

2− q3< r0.

Take q := min{q1, q2, q3}. Then q ∈ (qω, 2),

(3.32)q

p0<

n+ μ

n,2q

p0<

n+ 1

n+

r0 − 1

pΦr0and

2

2− q< r0.

By the third inequality in (3.32), we know that ω ∈ RH2/(2−q)(Rn). Moreover,

similar to the proof of [87, (3.51)], we know that

‖α‖L2(Rn) � ‖a‖T 22 (R

n×(0,R0]).

Then from this, ω ∈ RH2/(2−q)(Rn), Holder’s inequality, (2.1), (2.3) and Lemma

2.2(iii), we infer that when k = 0, there exists a positive constant C5 such that

‖M0‖Lqω(Rn) ≤ ‖α‖Lq

ω(2Q) + |m0|[ω(2Q)]

1q

|2Q|(3.33)

≤ ‖α‖Lqω(2Q) +

{∫2Q

|α(x)|qω(x) dx} 1

q

×{∫

2Q

[ω(x)]−q′q dx

} 1q′ [ω(2Q)]

1q

|2Q|

� ‖α‖Lqω(2Q) � ‖α‖L2(2Q)

{∫2Q

[ω(x)]2

2−q dx

} 1q−

12

� ‖a‖T 22 (R

n×(0,R0])

[ω(2Q)]1q

|2Q| 12≤ C5[ω(2Q)]

1q−1[ρ(ω(2Q))]−1,



which, together with (3.31) and supp (M0) ⊂ 2Q, implies thatM0/C5 is a (ρ, q, 0)ω-atom. When k ≥ 1, similar to the proof of [87, (3.53)], we know that for allx ∈ Rk(Q),

(3.34) |α(x)| � 2−k(n+1)|Q|− 12 ‖a‖T 2

2 (Rn×(0,R0]).

Thus, for all k ∈ N, from (3.34), Holder’s inequality, (2.1), Lemma 2.2(iii), theupper type 1

p0− 1 property of ρ and the first inequality of (3.32), it follows that

there exists a positive constant C6 such that

‖Mk‖Lqω(Rn) ≤ ‖α‖Lq

ω(Rk(Q)) + |mk|[ω(Rk(Q))]

1q

|Rk(Q)|(3.35)

≤ ‖α‖Lqω(Rk(Q))

+ ‖α‖Lqω(Rk(Q))

{∫Rk(Q)

[ω(x)]−q′/q dx

} 1q′

[ω(Rk(Q))]1q

|Rk(Q)|

� ‖α‖Lqω(Rk(Q)) � 2−k(n+1)[ω(Q)ρ(ω(Q))]−1[ω(2k+1Q)]

1q

� 2−k(n+1)2knq[ω(2k+1Q)]1q−1[ρ(2−knqω(2k+1Q))]−1

� C62−kn(n+1

n − qp0

)[ω(2k+1Q)]1q−1[ρ(ω(2k+1Q))]−1,

which, together with (3.31) and the fact that for each k ∈ N, supp (Mk) ⊂ 2k+1Q,

implies that for each k ∈ N, 2kn(n+1n − q

p0)Mk/C6 is a (ρ, q, 0)ω-atom. Moreover, by

an argument similar to that used in [87, p. 44], we know that∑∞

k=0 Mk convergesin L2(Rn). Let

λ1, k := C62−kn(n+1

n − qp0

) and a1, k := 2kn(n+1n − q

p0)Mk/C6

when k ∈ N, λ1, 0 := C5 and a1, 0 := M0/C5. Then {a1, k}∞k=0 is a sequence of(ρ, q, 0)ω-atoms. Furthermore, by the definitions of {λ1, k}∞k=0,

ω ∈ Aq(Rn) ∩RHr0(R

n),

(iii) and (v) of Lemma 2.2, the lower type 1pΦ

− 1 property of ρ and the second

inequality in (3.32), we know that for all λ ∈ (0,∞),

∞∑k=0

ω(2k+1Q)Φ

(|λ1, k|

λω(2k+1Q)ρ(ω(2k+1Q))

)(3.36)

�∞∑k=0

2(k+1)nqω(Q)Φ

(2−kn(n+1

n − qp0

)


)

�∞∑k=0

2(k+1)nqω(Q)Φ

⎛⎝2−kn(n+1n − q

p0)2

− kn(r0−1)pΦr0

λω(Q)ρ(ω(Q))

⎞⎠�

∞∑k=0

2−kn[

(n+1)p0n +

p0(r0−1)pΦr0

−2q]ω(Q)Φ

(1

λω(Q)ρ(ω(Q))

)� ω(Q)Φ

(1

λω(Q)ρ(ω(Q))

),



which, together with Lemma 3.7, implies that∞∑k=0

Mk ∈ hΦω(R

n).

To deal with the second sum in (3.32), by Holder’s inequality,

ω ∈ Aq(Rn) ∩RH 2

2−q(Rn),

Lemma 2.2(v), (3.34), |χk+1 − χk| � |2kQ|−1, Lemma 2.2(iii) and the upper type1p0

− 1 property of ρ, we know that there exists a positive constant C7 such that

for all k ∈ Z+,

‖Nk+1(χk+1 − χk)‖Lqω(Rn)(3.37)

≤ ‖Nk+1(χk+1 − χk)‖L2(2k+1Q)

{∫2k+1Q

[ω(x)]2

2−q dx

} 1q−

12

≤ |2k+1Q|− 12 |Nk+1|

[ω(2k+1Q)]1q

|2k+1Q| 12

� |2k+1Q|− 12

⎛⎝ ∞∑j=k+1

2−j

⎞⎠ |Q| 12 ‖a‖T 22 (R

n×(0,R0])

[ω(2k+1Q)]1q

|2k+1Q| 12

� 2−k(n+1) [ω(2k+1Q)]

1q

ω(Q)ρ(ω(Q))� 2−k(n+1)2knq

[ω(2k+1Q)]1q

ω(2k+1Q)ρ(2−knqω(2k+1Q))

� C72−kn(n+1

n − qp0

)[ω(2k+1Q)]1q−1[ρ(ω(2k+1Q))]−1.

This, combined with ∫Rn

[χk+1(x)− χk(x)] dx = 0

and supp (χk+1 − χk) ⊂ 2k+1Q, yields that for each k ∈ Z+,

2−kn(n+1n − q

p0)Nk+1(χk+1 − χk)/C7

is a (ρ, q, 0)ω-atom. Moreover, by an argument similar to that used in [87, p. 44],we know that

∞∑k=0

Nk+1(χk+1 − χk)

converges in L2(Rn). For all k ∈ Z+, let

λ2, k := C72−kn(n+1

n − qp0

) and a2, k := 2kn(n+1n − q

p0)Nk+1(χk+1 − χk)/C7.

Then {a2, k}∞k=0 is a sequence of (ρ, q, 0)ω-atoms. Similar to the proof of (3.36),we also see that for all λ ∈ (0,∞),

(3.38)

∞∑k=0

ω(2k+1Q)Φ

(|λ2, k|


)� ω(Q)Φ

(1

λω(Q)ρ(ω(Q))

),

which, together with Lemma 3.7, implies that∞∑k=0

Nk+1(χk+1 − χk) ∈ hΦω(R

n).



Let f ∈ L2(Rn) satisfy ‖Sloch,R0

(f)‖LΦω(Rn) < ∞. It is easy to see that for all

z ∈ C satisfying z �= 0 and | arg z| ∈ (0, π/2),

8

∫ R0

0

(t2ze−t2z)(t2ze−t2z)dt

t+ (2R2

0z + 1)e−2R20z = 1,

which, together with the H∞-functional calculus for L (see, for example, [62]),implies that

f = 8

∫ R0

0

(t2Le−t2L)(t2Le−t2L)(f)dt

t+[2R2

0Le−2R2

0L(f) + e−2R20L(f)

](3.39)

=: f1 + f2.

From the assumption ‖Sloch,R0

(f)‖LΦω(Rn) < ∞, we deduce that

t2Le−t2L(f) ∈ T loc, R0

Φ, ω (Rn)

and ∥∥Sloch,R0

(f)∥∥LΦ

ω(Rn)=∥∥∥t2Le−t2L(f)

∥∥∥T

loc, R0Φ, ω (Rn)

.

Then by Lemma 3.14, we know that there exist {μj}j ⊂ C and a sequence {aj}j of

T loc, R0

Φ, ω (Rn)-atoms such that for almost every (x, t) ∈ Rn × (0, R0],

t2Le−t2L(f)(x) =∑j

μjaj(x, t).(3.40)

For each j, let

αj := 8

∫ R0

0

t2Le−t2L(aj)dt

t.

Then by (3.39) and (3.40), similar to the proof of [49, Proposition 4.2], we see that

(3.41) f1 =∑j

μjαj

in L2(Rn). Replacing α in (3.28) by αj , consequently, we then denote Mk, Nk andχk in (3.30), λ1, k, λ2, k, a1, k and a2, k, respectively, by Mj, k, Nj, k, χj, k, λ1, j, k,λ2, j, k, a1, j, k and a2, j, k. Repeating the above procedure, we obtain

f1 =∑j

∞∑k=0

μjMj, k +∑j

∞∑k=0

μjNj, k+1(χj, k+1 − χj, k)

=:∑j

∞∑k=0

μjλ1, j, ka1, j, k +∑j

∞∑k=0

μjλ2, j, ka2, j, k,

where for each j,

{λ1, j, k}k∈Z+∪ {λ1, j, k}k∈Z+

⊂ C

and, {a1, j, k}k∈Z+and {a2, j, k}k∈Z+

are sequences of (ρ, q, 0)ω-atoms and bothsummations hold in L2(Rn), and hence in D′(Rn). Moreover, from (3.36) with Qand λ1, k replaced by Qj and λ1, j, k, (3.38) with Q and λ2, k replaced by Qj andλ2, j, k, and Lemma 3.14, we deduce that

Λ ({μjλ1, j, ka1, j, k}j, k) + Λ ({μjλ1, j, ka1, j, k)}j, k) � Λ ({μjaj}j)�∥∥Sloc

h,R0(f)

∥∥LΦ

ω(Rn).



This, combined with Lemma 3.7, implies that f1 ∈ hΦω (R

n) and

(3.42) ‖f1‖hΦω(Rn) �

∥∥Sloch,R0

(f)∥∥LΦ

ω(Rn).

Now we deal with f2. Denote by KR0the kernel of (2R2

0L+ 1)e−R20L. Then by

the mean value theorem for integrals, we know that for all x ∈ Rn,

f2(x) =

∫Rn

KR0(x, y)e−R2

0L(f)(y) dy =∑

Qk∈Q

∫Qk

KR0(x, y)e−R2

0L(f)(y) dy

=∑

Qk∈Q|Qk|mQk

(e−R2

0L(f))KR0

(x, yk),

where Q denotes the set of all unit cubes of Rn whose interiors are disjoint, andfor each k ∈ N, yk ∈ Qk may depend on x. For each k, we have

KR0(x, yk) =

∞∑i=0

KR0(x, yk)χSi(Qk) =:

∞∑i=0

Hk, i,

where S0(Qk) := 2Qk and for each i ∈ N, Si(Qk) := 2i+1Qk \ 2iQk. For each k,from (2.8), it follows that

(3.43)∣∣∣KR0

(x, yk)∣∣∣ � 1

(1 + |x− yk|)n+1.

By this, we conclude that there exists a positive constant C8 such that

‖Hk, 0‖Lqω(Rn) � {C8ω(2Qk)ρ(ω(2Qk))}[ω(2Qk)]

1q−1[ρ(ω(2Qk))]

−1.(3.44)

Thus, {C8ω(2Qk)ρ(ω(2Qk))}−1Hk, 0 is a (ρ, q, 0)ω-atom. For all i ∈ N, from (3.43),it follows that there exists a positive constant C9 such that

‖Hk, i‖Lqω(Rn) �

{∫Si(Qk)

ω(x)

[2il(Qk)](n+1)qdx

} 1q

(3.45)

�{C9|2iQk|−

n+1n ω

(2i+1Qk

)ρ(ω(2i+1Qk

))}×[ω(2i+1Qk

)] 1q−1 [

ρ(ω(2i+1Qk

))]−1,

which implies that C−19 |2iQk|

n+1n [ω(2i+1Qk)ρ(ω(2

i+1Qk))]−1Hk, i is a (ρ, q, 0)ω-

atom. Let

λ3, k, i := C9|Qk|mQk

(e−R2

0L(f))|2iQk|−

n+1n ω

(2i+1Qk

)ρ(ω(2i+1Qk)

)and

a3, k, i := C−19 |2iQk|

n+1n

[ω(2i+1Qk

)ρ(ω(2i+1Qk

))]−1Hk, i

for i ∈ N,

λ3, k, 0 := C8|Qk|mQk

(e−R2

0L(f))ω(2Qk)ρ(ω(2Qk))

and

a3, k, 0 := {C8ω(2Qk)ρ(ω(2Qk))}−1Hk, 0.

Then

f2 =∑k

∞∑i=0

λ3, k, ia3, k, i



and {a3, k, i}k, i∈Z+is a sequence of (ρ, q, 0)ω-atoms. From this, (3.44), (3.45),

l(Qk) = 1, the lower type p0 property of Φ and the first inequality in (3.32), wededuce that for all λ ∈ (0,∞),∑

Qk∈Q

∞∑i=0

ω(2i+1Qk)Φ

(|λ3, k, i|

λω(2i+1Qk)ρ(ω(2i+1Qk))

)

�∑

Qk∈Q

∞∑i=0

ω(2i+1Qk)Φ

(2−i(n+1)mQk

(|e−R20L(f)|)

λ

)

�∑

Qk∈Q

∞∑i=0

2−inq2−(n+1)p0ω(Qk)Φ

(mQk

(|e−R20L(f)|)

λ

)

�∑

Qk∈Qω(Qk)Φ

(mQk

(|e−R20L(f)|)

λ

),

which, together with Lemma 3.7, implies that f2 ∈ hΦω (R

n) and

(3.46) ‖f2‖hΦω(Rn) � inf

⎧⎨⎩λ ∈ (0,∞) :∑

Qk∈Qω(Qk)Φ

(mQk

(|e−R20L(f)|)

λ

)≤ 1

⎫⎬⎭ .

From this, (3.42) and (3.46), we infer that f ∈ hΦω (R

n) and

‖f‖hΦω(Rn) �

∥∥Sloch,R0

(f)∥∥LΦ

ω(Rn)

+ inf

⎧⎨⎩λ ∈ (0,∞) :∑

Qk∈Qω(Qk)Φ

(mQk

(|e−R20L(f)|)

λ

)≤ 1

⎫⎬⎭ ,

which completes the proof of Proposition 3.13(i).Now we prove Proposition 3.13(ii). Let f ∈ L2(Ω) satisfy ‖Sloc

h,R0(f)‖LΦ

ω(Ω) < ∞.

Similar to the proof of (3.39), we know that (3.39) also holds in this case. Let f1and f2 be as in (3.39).

We first deal with f1. Similar to the proof of (3.41), we know that

f1 =∑j

μjαj

in L2(Ω), where {μj}j ⊂ C and for each j,

αj := 8

∫ R0

0

t2Le−t2L(aj)dt

t

and aj is a T loc, R0

Φ, ω (Ω)-atom. For any T loc, R0

Φ, ω (Ω)-atom a supported in Q ∩ Ω, let

α := 8

∫ R0

0

t2Le−t2L(a)dt

t.

To show f1 ∈ hΦω, r(Ω), it suffices to show that there exist a function α on Rn, a

sequence {λi}i of numbers and a sequence {bi}i of (ρ, q, 0)ω-atoms such that

(3.47) α|Ω = α,

α =∑i

λibi in L2(Rn)



and for all λ ∈ (0,∞),

(3.48)∑i

ω(Qi)Φ

(|λi|

λω(Qi)ρ(ω(Qi))

)� ω(Q ∩ Ω)Φ

(1

λω(Q ∩ Ω)ρ(ω(Q ∩ Ω))

),

where for each i, supp (bi) ⊂ Qi and Q ∩ Ω appears in the support of a. Indeed,if (3.47) and (3.48) hold, then by (3.47), we know that for each j, there exists afunction αj on Rn such that αj |Ω = αj . Let

f1 :=∑j

μjαj .

Then f1|Ω = f1. Furthermore, from (3.48), we deduce that there exist {λj, i}j, i ⊂ C

and a sequence {bj, i}j, i of (ρ, q, 0)ω-atoms such that

f1 =∑j

∑i

μjλj, ibj, i

and for all λ ∈ (0,∞),∑j, i

ω(Qj, i)Φ

(|μjλj, i|

λω(Qj, i)ρ(ω(Qj, i))

)

�∑j

ω(Qj ∩ Ω)Φ

(|μj |

λω(Qj ∩ Ω)ρ(ω(Qj ∩ Ω))

),

which, together with Lemmas 3.7 and 3.14, implies that∥∥∥f1∥∥∥hΦω(Rn)

∼∥∥∥f1∥∥∥

hρ, q, 0ω (Rn)

�∥∥Sloc

h,R0(f)

∥∥LΦ

ω(Ω).

From this and Definition 1.3, we deduce that f1 ∈ hΦω, r(Ω) and

(3.49) ‖f1‖hΦω, r(Ω) �

∥∥Sloch,R0

(f)∥∥LΦ

ω(Ω).

Let Q := Q(x0, r0). Now we show (3.47) and (3.48) by considering the followingtwo cases for Q which appears in the support of a.

Case 1) 8Q ∩ Ω� �= ∅. In this case, let

Rk(Q) := (2k+1Q \ 2kQ) ∩ Ω

if k ≥ 3 and R0(Q) := 8Q ∩ Ω. Let

JΩ := {k ∈ N : k ≥ 3, |Rk(Q)| > 0}.For k ∈ JΩ ∪ {0}, let χk := χRk(Q), χk := |Rk(Q)|−1χk and

mk :=

∫Rk(Q)

α(x) dx.

Then we have

α = αχ0 +∑k∈JΩ

αχk

almost everywhere and also in L2(Ω). Take the cube Q ⊂ Rn such that the center

of Q, x˜Q ∈ Ω�, l(Q) = l(Q) and dist (Q, Q) ∼ l(Q). Then there exists a cube Q∗

0

such that (8Q ∪ Q) ⊂ Q∗0 and

(3.50) l(Q∗0) ∼ l(Q).



Let

H0 := αχ0 −1

|Q ∩ Ω�|

{∫R0(Q)

α(x) dx

}χ

˜Q∩Ω� .

Then ∫Rn

H0(x) dx = 0

and supp (H0) ⊂ Q∗0. Similar to the proof of [87, (3.51)], we conclude that

(3.51) ‖α‖L2(Ω) � ‖a‖T 22 (Ω×(0,R0]).

By the assumption that Ω� is an unbounded strongly Lipschitz domain and Lemma

3.10, we know that |Q ∩ Ω�| ∼ |Q|. From this, Holder’s inequality,

ω ∈ Aq(Rn) ∩RH2/(2−q)(R

n),

(2.3), (3.50), (3.51) and, (iii) and (v) of Lemma 2.2, it follows that there exists apositive constant C10 such that

‖H0‖Lqω(Rn) ≤ ‖H0‖L2(Q∗

0)

{∫Q∗

0

[ω(x)]2/(2−q) dx

} 1q−

12

� ‖H0‖L2(Q∗0)[ω(Q∗

0)]1q

|Q∗0|

12

�

⎧⎨⎩‖α‖L2(Ω) +1

|Q ∩ Ω�| 12

(∫R0(Q)

|α(x)|2 dx) 1

2

|Q ∩ Ω| 12

⎫⎬⎭ [ω(Q∗0)]

1q

|Q∗0|

12

� [ω(Q∗0)]

1q

|Q∗0|

12

‖α‖L2(Ω) �[ω(Q∗

0)]1q

|Q∗0|

12

‖a‖T 22 (Ω×(0,R0])

� [ω(Q∗0)]

1q |Q ∩ Ω| 12

|Q∗0|

12ω(Q ∩ Ω)ρ(ω(Q ∩ Ω))

� C10[ω(Q∗0)]

1q−1[ρ(ω(Q∗

0))]−1.

Let λ0 := C10 and b0 := H0/C10. Then H0 = λ0b0 and b0 is a (ρ, q, 0)ω-atom.To finish the proof in this case, we need the following Fact 1, whose proof is

similar to the usual Whitney decomposition of an open set in Rn; see, for example,[76]. We omit the details.

Fact 1. For all k ∈ JΩ, there exists the Whitney decomposition {Qk, i}i of Rk(Q)about ∂Ω, where {Qk, i}i are dyadic cubes of Rn with disjoint interiors and for eachi, 2Qk, i ⊂ Ω but 4Qk, i ∩ ∂Ω �= ∅.

Let {Qk, i}k∈JΩ, i be as in Fact 1. Then for each k ∈ JΩ,

αχRk(Q) =∑i

αχQk, i

almost everywhere. Similar to the proof of [87, (3.53)], we know that for all x ∈Rk(Q),

|α(x)| � 2−k(n+1)|Q ∩ Ω|− 12 ‖a‖T 2

2 (Ω×(0,R0]).(3.52)



Moreover, by Lemma 3.15, we see that for each k and i, there exists a cube Qk, i ⊂Ω� such that l(Qk, i) = l(Qk, i) and

dist (Qk, i, Qk, i) ∼ l(Qk, i).

Then for each k and i, there exists a cube Q∗k, i such that (Qk, i ∪ Qk, i) ⊂ Q∗

k, i and

l(Q∗k, i) ∼ l(Qk, i). For each k and i, let

Hk, i := αχQk, i− 1

|Qk, i|

{∫Qk, i

α(x) dx

}χ

˜Qk, i.

Then ∫Rn

Hk, i(x) dx = 0 and supp (Hk, i) ⊂ Q∗k, i.

Furthermore, from Holder’s inequality, ω ∈ RH2/(2−q)(Rn), (2.3), (3.52) and

l(Q∗k, i) ∼ l(Qk, i),

we infer that there exists a positive constant C11 such that

‖Hk, i‖Lqω(Rn) ≤ ‖α‖L2(Q∗

k, i)

{∫Q∗

k, i

[ω(x)]2/(2−q) dx

} 1q−

12

(3.53)

� 2−k(n+1)|Q ∩ Ω|− 12 |Q∗

k, i|12 ‖a‖T 2

2 (Ω×(0,R0])

[ω(Q∗k, i)]

1q

|Q∗k, i|

12

� C112−k(n+1)[ω(Q∗

k, i)]1q [ω(Q ∩ Ω)ρ(ω(Q ∩ Ω))]−1.

For each k and j, let

λk, j :=C112

−k(n+1)ω(Q∗k, i)ρ(ω(Q

∗k, i))

ω(Q ∩ Ω)ρ(ω(Q ∩ Ω))

and

bk, j :=2k(n+1)Hk, jω(Q ∩ Ω)ρ(ω(Q ∩ Ω))

C11ω(Q∗k, i)ρ(ω(Q

∗k, i))

.

Then for each k and j, bk, j is a (ρ, q, 0)ω-atom and Hk, j := λk, jbk, j . Let

α := H0 +∑k∈JΩ

∑i

Hk, i = λ0b0 +∑k∈JΩ

∑i

λk, ibk, j .

Then by the constructions of H0 and {Hk, i}k∈JΩ, i, we know that α|Ω = α. Similarto [87, (3.55)], we know that

α = λ0b0 +∑k∈JΩ

∑i

λk, ibk, j

in L2(Rn). Moreover, by the definitions of λ0 and λk, i, the lower type p0 property ofΦ, Lemma 2.2(iii) and the first inequality in (3.32), we know that for all λ ∈ (0,∞),

ω(Q∗0)Φ

(λ0

λω(Q∗0)ρ(ω(Q

∗0))

)+

∑k∈JΩ

∑i

ω(Q∗k, i)Φ

(λk, i

λω(Q∗k, i)ρ(ω(Q

∗k, i))

)(3.54)

� ω(Q ∩ Ω)Φ

(1


)



+∑k∈JΩ

∑i

ω(Qk, i)Φ

(2−k(n+1)


)

� ω(Q ∩ Ω)Φ

(1


)+

∞∑k=3

ω(2k+1Q ∩ Ω)Φ

(2−k(n+1)


)

� ω(Q ∩ Ω)Φ

(1


){1 +

∞∑k=3

2−[k(n+1)p0−kqn]

}

� ω(Q ∩ Ω)Φ

(1

ω(Q ∩ Ω)ρ(ω(Q ∩ Ω))

),

which implies that (3.48) holds in Case 1).

Case 2) 8Q ⊂ Ω. In this case, let k0 ∈ N be such that 2k0Q ⊂ Ω but

2k0+1Q ∩ ∂Ω �= ∅.Then k0 ≥ 3. Let

Rk(Q) := (2k+1Q \ 2kQ) ∩ Ω

for k ∈ N, and R0(Q) := 2Q. Let

JΩ, k0:= {k ∈ N : k ≥ k0 + 1, |Rk(Q)| > 0}.

For k ∈ Z+, let χk := χRk(Q), χk := |Rk(Q)|−1χk,

mk :=

∫Rk(Q)

α(x) dx, Mk := αχk −mkχk

and Mk := αχk. Then

α =

k0∑k=0

Mk +∑

k∈JΩ, k0

Mk +

k0∑k=0

mkχk.

For k ∈ {0, 1, · · · , k0}, by the definition of Mk, we know that∫Rn

Mk(x) dx = 0

and supp (Mk) ⊂ 2k+1Q. Moreover, similar to the estimates of (3.33) and (3.35),we see that there exists a positive constant C12 such that for all k ∈ {0, 1, · · · , k0},

‖Mk‖Lqω(Rn) ≤ C122

−kn(n+1n − q

p0) [ω (2k+1Q

)] 1q−1 [

ρ(ω(2k+1Q

))]−1.

For each k ∈ {0, · · · , k0}, let

λ1, k := C122−kn(n+1

n − qp0

) and b1, k := C−112 2kn(

n+1n − q

p0)Mk.

Thus, for each k ∈ {0, · · · , k0}, b1, k is a (ρ, q, 0)ω-atom and Mk = λ1, kb1, k.Moreover, similar to the estimate of (3.36), we know that for all λ ∈ (0,∞),

k0∑k=0

ω(2k+1Q)Φ

(λ1, k


)(3.55)

� ω(Q)Φ

(1

λω(Q)ρ(ω(Q))

).



For each k ∈ JΩ, k0, by Fact 1, there exists the Whitney decomposition {Qk, i}i of

Rk(Q) about ∂Ω such that⋃

i Qk, i = Rk(Q) and for each i, Qi satisfies 2Qk, i ⊂ Ωand 4Qk, i ∩ ∂Ω �= ∅. Then

Mk =∑i

αχQk, i

almost everywhere. Moreover, by Lemma 3.15, for each k and i, there exists a cube

Qk, i ⊂ Ω� such that l(Qk, i) = l(Qk, i) and dist (Qk, i, Qk, i) ∼ l(Qk, i). Then for

each k and i, there exists a cube Q∗k, i such that Qk, i ∪ Qk, i ⊂ Q∗

k, i and l(Q∗k, i) ∼

l(Qk, i). For each k and i, let

Hk, i := αχQk, i− 1

|Qk, i|

{∫Qk, i

α(x) dx

}χ

˜Qk, i.

Then ∫Rn

Hk, i(x) dx = 0

and supp (Hk, i) ⊂ Q∗k, i. Furthermore, similar to the proof of (3.53), we conclude

that there exists a positive constant C13 such that for each k ∈ JΩ, k0and i,

‖Hk, i‖Lqω(Rn) ≤ C132

−k(n+1)[ω(Q∗k, i)]

1q [ω(Q ∩ Ω)ρ(ω(Q ∩ Ω))]−1.

For each k and j, let

λk, j :=C132

−k(n+1)ω(Q∗k, i)ρ(ω(Q

∗k, i))

ω(Q ∩ Ω)ρ(ω(Q ∩ Ω))

and

bk, j :=2k(n+1)Hk, jω(Q ∩ Ω)ρ(ω(Q ∩ Ω))

C13ω(Q∗k, i)ρ(ω(Q

∗k, i))

.

Then for each k and j, bk, j is a (ρ, q, 0)ω-atom and Hk, j := λk, jbk, j . Furthermore,similar to the proof of (3.54), we see that for all λ ∈ (0,∞),∑

k∈JΩ, k0

∑i

ω(Q∗k, i)Φ

(λk, i

λω(Q∗k, i)ρ(ω(Q

∗k, i))

)(3.56)

� ω(Q ∩ Ω)Φ

(1


).

For j ∈ {0, 1, · · · , k0}, let

Nj :=

k0∑k=j

mk.

It is easy to see that

k0∑k=0

mkχk =

k0∑k=1

(χk − χk−1)Nk +N0χ0.

Similar to the proof of (3.37), we know that there exists a positive constant C14

such that for each k ∈ {1, 2, · · · , k0},‖Nk(χk − χk−1)‖Lq

ω(Rn)(3.57)

≤ C142−kn(n+1

n − qp0

) [ω (2kQ)] 1q−1 [

ρ(ω(2kQ

))]−1.



For all k ∈ {1, 2, · · · , k0}, let

λ2, k := C142−kn(n+1

n − qp0

) and b2, k := C−114 2kn(

n+1n − q

p0)Nk(χk − χk−1).

Then from (3.57), ∫Rn

[χk(x)− χk−1(x)] dx = 0

and supp (χk − χk−1) ⊂ 2kQ, we deduce that for each k ∈ {1, 2, · · · , k0}, b2, k is a(ρ, q, 0)ω-atom. Similar to the proof of (3.36), we know that for all λ ∈ (0,∞),

(3.58)

k0∑k=1

ω(2kQ)Φ

(λ2, k

λω(2kQ)ρ(ω(2kQ))

)� ω(Q)Φ

(1

λω(Q)ρ(ω(Q))

).

Finally we deal with N0χ0. By

2k0−1r0 < dist (x0, ∂Ω) ≤ 2k0r0,

we conclude that there exist a positive integer M and a sequence {Q0, i}Mi=1 of cubessuch that

(i) M ∼ 2k0 ;(ii) for all i ∈ {1, 2, · · · , M}, l(Q0, i) = 2r0 and Q0, i ⊂ Ω;(iii) for all i ∈ {1, 2, · · · , M − 1}, Q0, i ∩Q0, i+1 �= ∅ and

dist (Q0, i, ∂Ω) ≥ dist (Q0, i+1, ∂Ω);

(iv) 2Q0,M ∩ ∂Ω �= ∅.Then by Lemma 3.15, there exists a cube Q0,M+1 ⊂ Ω� such that l(Q0,M+1) = r0and

dist (Q0,M , Q0,M+1) ∼ r0.

Let

H0, 1 := N0χ0 −N0

|Q0, 1|χQ0, 1

and

H0, i :=N0

|Q0, i−1|χQ0, i−1

− N0

|Q0, i|χQ0, i

with i ∈ {2, · · · , M+1}. Obviously, for all i ∈ {1, 2, · · · , M+1}, by the definitionof H0, i, we see that ∫

Rn

H0, i(x) dx = 0

and there exists a cube Q∗0, i ⊂ Rn such that supp (H0, i) ⊂ Q∗

0, i and

l(Q∗0, i) ∼ l(Q).(3.59)

Similar to the proof of [87, (3.66)], we know that

|N0| � 2−k0(n+1)/n|Q| 12 ‖a‖T 22 (Ω×(0,R0]).(3.60)

For each i ∈ {1, 2, · · · , M+1}, from Holder’s inequality, ω ∈ RH 22−q

(Rn), (2.3), the

definition of H0, i, (3.59) and (3.60), it follows that there exists a positive constantC15 such that

‖H0, i‖Lqω(Rn) ≤ ‖H0, i‖L2(Rn)

{∫Q∗

0, i

[ω(x)]2

2−q dx

} 1q−

12

(3.61)



� |N0||Q|− 12[ω(Q∗

0, i)]1q

|Q∗0, i|

12

� 2−k0(n+1)/n[ω(Q∗

0, i)]1q

|Q∗0, i|

12

‖a‖T 22 (Ω×(0,R0])

� C152−k0(n+1)/n[ω(Q∗

0, i)]1q [ω(Q)ρ(ω(Q))]−1.

For each i ∈ {1, · · · , M + 1}, let

λ3, i := C152−k0(n+1)/n

ω(Q∗0, i)ρ(ω(Q

∗0, i))

ω(Q)ρ(ω(Q))

and

b3, i :=2k0(n+1)/nω(Q)ρ(ω(Q))H0, i

C15ω(Q∗0, i)ρ(ω(Q

∗0, i))

.

Then for any i ∈ {1, · · · , M + 1}, from (3.61),∫Rn

H0, i(x) dx = 0

and supp (H0, i) ⊂ Q∗0, i, we infer that b3, i is a (ρ, q, 0)ω-atom. By the construction

of {Q0, i}M+1i=1 , we know that there exists a positive constant C16 such that

M+1⋃i=1

Q0, i ⊂ C162k0Q.

By this, the lower p0 property of Φ, Lemma 2.2(iii) and the first inequality in (3.32),we conclude that for all λ ∈ (0,∞),

M+1∑i=1

ω(Q∗0, i)Φ

(λ3, i

λω(Q∗0, i)ρ(ω(Q

∗0, i))

)(3.62)

�M+1∑i=1

ω(Q0, i)Φ

(2−k0(n+1)/n

λω(Q)ρ(ω(Q))

)� ω(C162

k0Q)2−k0(n+1)p0

n Φ

(1

λω(Q)ρ(ω(Q))

)� 2k0[nq−(n+1)p0/n]ω(Q)Φ

(1

λω(Q)ρ(ω(Q))

)� ω(Q)Φ

(1

λω(Q)ρ(ω(Q))

).

Let

α :=

k0∑i=1

Mk +∑

k∈JΩ, k0

∑i

bk, i +

k0∑k=1

(χk − χk−1)Nk +

M+1∑i=1

H0, i.

Similar to the proof of [87, (3.55)], we see that the above equality holds in L2(Rn).By the definition of α, we know that α|Ω = α. Furthermore, from (3.55), (3.56),(3.58) and (3.62), it follows that α ∈ hΦ

ω(Rn) and (3.48) holds in Case 2).



Finally, we deal with f2. Denote by KR0the kernel of (2R2

0 + 1)Le−R20L. Then

by the mean value theorem for integrals, we know that

f2 =

∫Ω

KR0(x, y)e−R2

0L(f)(y) dy

=∑

Qk∈Q, Qk∩Ω �=∅

∫Qk∩Ω

KR0(x, y)e−R2

0L(f)(y) dy

=∑

Qk∈Q, Qk∩Ω �=∅|Qk ∩ Ω|mQk∩Ω

(e−R2

0L(f))KR0

(x, yk),

where for each k ∈ N, yk ∈ Qk ∩ Ω may depend on x. For each k, we have

KR0(x, yk) =

∞∑i=0

KR0(x, yk)χSi(Qk) :=

∞∑i=0

Hk, i,

where S0(Qk) := 2Qk ∩ Ω and for each i ∈ N,

Si(Qk) := (2i+1Qk \ 2iQk) ∩ Ω.

For each k, by (2.8), we see that for all x ∈ Ω,

(3.63)∣∣∣KR0

(x, yk)∣∣∣ � 1

(1 + |x− yk|)n+1.

From this, we infer that there exists a positive constant C17 such that

‖Hk, 0‖Lqω(Rn) � [ω(2Qk)]

1q � C17ω(2Qk)ρ(ω(2Qk))[ω(2Qk)]

1q−1[ρ(ω(2Qk))]

−1.

Thus, {C17ω(2Qk)ρ(ω(2Qk))}−1bk, 0 is a (ρ, q, 0)ω-atom. For all i ∈ N, by (3.63),we conclude that there exists a positive constant C18 such that

‖Hk, i‖Lqω(Rn) �

{∫Si(Qk)

ω(x)

[2il(Qk)](n+1)qdx

} 1q

(3.64)

� |2iQk|−n+1n

[ω(2i+1Qk

)]1/q�{C18|2iQk|−

n+1n ω

(2i+1Qk

)ρ(ω(2i+1Qk

))}×[ω(2i+1Qk

)] 1q−1 [

ρ(ω(2i+1Qk

))]−1,

which implies that

|2iQk|(n+1)/n[ω(2i+1Qk

)ρ(ω(2i+1Qk

))]−1Hk, i/C18

is a (ρ, q, 0)ω-atom. Let

λ3, k, i := C18|Qk ∩ Ω|mQk∩Ω

(e−R2

0L(f))|2iQk|−(n+1)/nω

(2i+1Qk

)ρ(ω(2i+1Qk

))and

a3, k, i := C−118 |2iQk|

n+1n

[ω(2i+1Qk

)ρ(ω(2i+1Qk

))]−1Hk, i

when i ∈ N,

λ3, k, 0 := C17|Qk ∩ Ω|mQk∩Ω

(e−R2

0L(f))ω(2Qk)ρ(ω(2Qk))

and

a3, k, 0 := {C17ω(2Qk)ρ(ω(2Qk))}−1Hk, 0.



Then

f2 =∑k

∞∑i=0

λ3, k, ia3, k, i

and {a3, k, i}k, i∈Z+is a sequence of (ρ, q, 0)ω-atoms. From this, (3.44), (3.45) and

l(Qk) = 1, Lemma 2.2(iii) and the first inequality in (3.32), we deduce that for allλ ∈ (0,∞), ∑


∞∑i=0

ω(2i+1Qk)Φ

(|λ3, k, i|

λω(2i+1Qk)ρ(ω(2i+1Qk))

)

�∑


∞∑i=0

ω(2i+1Qk)Φ

(2−i(n+1)mQk∩Ω(|e−R2

0L(f)|)λ

)

�∑

˜Qk∈QΩ

∞∑i=0

2−inq2−(n+1)p0ω(Qk

)Φ

(m

˜Qk∩Ω(|e−R20L(f)|)

λ

)

�∑

˜Qk∈QΩ

ω(Qk ∩ Ω

)Φ

(m

˜Qk∩Ω(|e−R20L(f)|)

λ

),

where for each k, Qk is as in Definition 3.3, which, together with Lemma 3.7, impliesthat f2 ∈ hΦ

ω, z(Ω) and hence f2 ∈ hΦω, r(Ω), and

‖f2‖hΦω, r(Ω) ≤ ‖f2‖hΦ

ω, z(Ω)(3.65)

� inf

⎧⎨⎩λ ∈ (0,∞) :∑

˜Qk∈QΩ

ω(Qk ∩ Ω

)

× Φ

(m

˜Qk∩Ω(|e−R20L(f)|)

λ

)≤ 1

⎫⎬⎭ .

From (3.39), (3.49) and (3.65), we infer that f ∈ hΦω, r(Ω) and

‖f‖hΦω, r(Ω) �

∥∥Sloch,R0

(f)∥∥LΦ

ω(Ω)+ inf

⎧⎨⎩λ ∈ (0,∞) :∑

˜Qk∈QΩ

ω(Qk ∩ Ω

)

× Φ

(m

˜Qk∩Ω(|e−R20L(f)|)

λ

)≤ 1

⎫⎬⎭ ,

which completes the proof of Proposition 3.13(ii).

Now we prove Proposition 3.13(iii). Let f ∈ L2(Ω) satisfy

‖Sh,R0(f)‖LΦ

ω(Ω) < ∞.

By the proof of (3.39), we know that (3.39) also holds in this case. Let f1 and f2be as in (3.39). Denote the zero extensions out of Ω of f1 and f2 respectively by

f1 and f2. Similar to the proof of f1 ∈ hΦω (R

n) in Proposition 3.13(i), we conclude



that f1 ∈ hΦω (R

n), and hence f1 ∈ hΦω, z(Ω) and

‖f1‖hΦω, z(Ω) =

∥∥∥f1∥∥∥hΦω(Rn)

�∥∥Sloc

h,R0(f)

∥∥LΦ

ω(Ω).

Similar to the proof of f2 ∈ hΦω (R

n) in Proposition 3.13(ii), we know that f2 ∈hΦω(R

n), and hence f2 ∈ hΦω, z(Ω) and

‖f2‖hΦω, z(Ω) =

∥∥∥f2∥∥∥hΦω(Rn)

� inf

⎧⎨⎩λ ∈ (0,∞) :∑

˜Qk∈QΩ

ω(Qk ∩ Ω

)

× Φ

(m

˜Qk∩Ω(|e−R20L(f)|)

λ

)≤ 1

⎫⎬⎭ .

Let f := f1+f2. Then f is the zero extension out of Ω of f . By the above argument,

we know that f ∈ hΦω (R

n) and hence f ∈ hΦω, z(Ω). Furthermore,

‖f‖hΦω, z(Ω) �

∥∥Sloch,R0

(f)∥∥LΦ

ω(Ω)+ inf

⎧⎨⎩λ ∈ (0,∞) :∑

˜Qk∈QΩ

ω(Qk ∩ Ω

)

× Φ

(m

˜Qk∩Ω(|e−R20L(f)|)

λ

)≤ 1

⎫⎬⎭ ,

which completes the proof of Proposition 3.13(iii) and hence Proposition 3.13. �

Now we prove Theorem 1.4 by using Propositions 3.4, 3.8, 3.12 and 3.13.

Proof of Theorem 1.4. We first show Theorem 1.4(i). Let f ∈ hΦω (R

n) ∩ L2(Rn),R0 ∈ [ 12 ,∞) and Q ∈ Q. Then

mQ

(|e−R2

0L(f)|)≤ inf

x∈QN loc, 2R0

h (f)(x),

which implies that for all λ ∈ (0,∞),∑Qk∈Q

ω(Qk)Φ

(mQk

(|e−R20L(f)|)

λ

)≤

∑Qk∈Q

∫Qk

Φ

(N loc, 2R0

h (f)(x)

λ

)ω(x) dx

�∫Rn

Φ

(N loc, 2R0

h (f)(x)

λ

)ω(x) dx.

From this, together with Propositions 3.4(i), 3.8, 3.12 and 3.13(i), we deduce that

‖f‖hΦω(Rn) ∼

∥∥∥N loc, 2R0

h (f)(x)∥∥∥LΦ

ω(Rn)

∼∥∥∥Sloc

h,R0(f)

∥∥∥LΦ

ω(Rn)

+ inf

⎧⎨⎩λ ∈ (0,∞) :∑

Qk∈Qω(Qk)Φ

(mQk

(|e−R20L(f)|)

λ

)≤ 1

⎫⎬⎭



∼∥∥Sloc

h,R0(f)

∥∥LΦ

ω(Rn)

+ inf

⎧⎨⎩λ ∈ (0,∞) :∑

Qk∈Qω(Qk)Φ

(mQk

(|e−R20L(f)|)

λ

)≤ 1

⎫⎬⎭ ,

which, together with the arbitrariness of R0 ∈ [ 12 ,∞) and f ∈ hΦω (R

n), implies that

hΦω (R

n) ∩ L2(Rn) = hΦNh, ω

(Rn) ∩ L2(Rn)

= hΦ˜Sh, ω

(Rn) ∩ L2(Rn) = hΦSh, ω

(Rn) ∩ L2(Rn)

with equivalent quasi-norms.To finish the proof of Theorem 1.4(i), we claim that hΦ

ω (Rn) ∩ L2(Rn) is dense

in hΦω (R

n). We now prove the claim. For any q ∈ (2qω,∞] and s ∈ Z+ satisfyings ≥ �n(qω/p−Φ − 1)�, denote the vector space of all finite linear combinations of(ρ, q, s)ω-atoms by hρ, q, s

ω,fin (Rn). By Lemma 3.7 and the definition of hρ, q, s

ω, fin (Rn), we

know that hρ, q, sω,fin (R

n) is a dense subspace of hΦω(R

n). For any f ∈ hρ, q, sω, fin (R

n), let

g :=N∑i=1

λiai,

where N ∈ N, and for each i ∈ {1, · · · , N}, λi ∈ C and ai is a (ρ, q, s)ω-atom. Foreach i ∈ {1, · · · , N}, let supp (ai) ⊂ Qi. By q ∈ (2qω,∞] and the definition of qω,we see that ω ∈ Aq/2(R

n). From this and Holder’s inequality, we deduce that foreach i ∈ {1, · · · , N},

‖ai‖L2(Rn) ≤ ‖ai‖Lqω(Rn)

{∫Qi

[ω(x)]−2

q−2

} 12−

1q

� [ω(Qi)]1q−1[ρ(ω(Qi))]

−1 |Qi|12

[ω(Qi)]1q

� |Qi|12

ω(Qi)ρ(ω(Qi)),

which implies that ai ∈ L2(Rn). By this and the definition of f , we conclude thatf ∈ hΦ

ω(Rn) ∩ L2(Rn), which, together with the fact that hρ, q, s

ω, fin (Rn) is dense in

hΦω(R

n), implies that hΦω (R

n) ∩ L2(Rn) is dense in hΦω (R

n). Thus, the claim holds.From this, (3.64), the fact that

hΦNh, ω

(Rn) ∩ L2(Rn), hΦ˜Sh, ω

(Rn) ∩ L2(Rn) and hΦSh, ω

(Rn) ∩ L2(Rn)

are, respectively, dense in hΦNh, ω

(Rn), hΦ˜Sh, ω

(Rn) and hΦSh, ω

(Rn), and a density

argument, we deduce that the spaces hΦω (R

n), hΦNh, ω

(Rn), hΦ˜Sh, ω

(Rn) and hΦSh, ω

(Rn)

coincide with equivalent quasi-norms. This finishes the proof of Theorem 1.4(i).From the definitions of the spaces hΦ

ω, r(Ω) and hΦω, z(Ω) and the fact that hΦ

ω (Rn)∩

L2(Rn) is dense in hΦω (R

n), we deduce that hΦω, r(Ω) ∩ L2(Ω) and hΦ

ω, z(Ω) ∩ L2(Ω)

are, respectively, dense in hΦω, r(Ω) and hΦ

ω, z(Ω). The remainder of the proofs of (ii)and (iii) of Theorem 1.4 is similar to that of Theorem 1.4(i). We omit the details.This finishes the proof of Theorem 1.4. �




In this section, we give the proof of Theorem 1.7.

Proof of Theorem 1.7. We borrow some ideas from [66, 65] and [84]. We proveTheorem 1.7 by using the following strategy: first, we show that (i) and (iv) areequivalent; then we prove the equivalence between (ii) and (iii); finally, we showthat (ii) implies (i), which, together with the standard proof of the implication(iv) =⇒ (iii), completes the proof of Theorem 1.7. Thus, we divide the whole proofinto the following four steps.

Step I. (i) ⇐⇒ (iv). First we prove that (i) implies (iv). Let f ∈ hΦω, r(Ω). Then

there exists F ∈ hΦω(R

n) such that F |Ω = f and

(4.1) ‖F‖hΦω(Rn) ∼ ‖f‖hΦ

ω, r(Ω).

From Lemma 3.7, it follows that there exist a sequence {ai}i of (ρ, ∞, s)ω-atomsand {λi}i ⊂ C such that F =

∑i λiai in D′(Rn) and

(4.2) ‖F‖hΦω(Rn) ∼ Λ ({λiai}i) ,

where Λ ({λiai}i) is as in Definition 3.6.For any (ρ, ∞, s)ω-atom a, let supp (a) ⊂ Q. Since f = F |Ω, we only need to

consider the case that Q ∩ Ω �= ∅. If 2Q ⊂ Ω and 4Q ∩ ∂Ω = ∅, then a is a type(a) local (ρ, ∞, s)ω-atom. If 2Q ⊂ Ω and 4Q ∩ ∂Ω �= ∅, then a is a type (b) local(ρ, ∞, s)ω-atom. If 2Q ∩ ∂Ω �= ∅, by the Whitney decomposition over Q ∩ Ω with∂Ω, we know that there exists a family of cubes, {Qj}j , with disjoint interiors suchthat 2Qj ⊂ Ω, 4Qj ∩ ∂Ω �= ∅ and

Q ∩ Ω =⋃j

Qj .

Thus, a|Ω =∑

j aχQj. For each j, let

bQj:=

ω(Q)ρ(ω(Q))

ω(Qj)ρ(ω(Qj))aχQj

and

μj :=ω(Qj)ρ(ω(Qj))

ω(Q)ρ(ω((Q)).

Then bQjis a type (b) local (ρ, ∞, s)ω-atom, a|Ω =

∑j μjbQj

and for all λ ∈ (0, ∞),∑j

ω(Qj)Φ

(μj

λω(Qj)ρ(ω(Qj))

)≤∑j

ω(Qj)Φ

(1

λω(Q)ρ(ω(Q))

)(4.3)

≤ ω(Q)Φ

(1

λω(Q)ρ(ω(Q))

).

Thus, from the above observation, we deduce that

f =∑k

μ1, kb1, k +∑j

μ2, jb2, j

in D′(Rn), where {b1, k}k is a sequence of type (a) local (ρ, ∞, s)ω-atoms, {b2, j}j asequence of type (b) local (ρ, ∞, s)ω-atoms, and {μ1, k}k ∪{μ2, j}j ⊂ C. Moreover,



by (4.3), we know that for all λ ∈ (0, ∞),∑k

ω(Q1, k)Φ

(|μ1, k|

λω(Q1, k)ρ(ω(Q1, k))

)+∑j

ω(Q2, j)Φ

(|μ2, j |

λω(Q2, j)ρ(ω(Q2, j))

)

≤∑i

ω(Qi)Φ

(|λi|

λω(Qi)ρ(ω(Qi))

),

where for each k, j and i, supp (b1, k) ⊂ Q1, k, supp (b2, j) ⊂ Q2, j and supp (ai) ⊂Qi. This, combined with (4.1) and (4.2), implies that

‖f‖hρ, ∞, sω (Ω) � ‖F‖hρ,∞, s

ω (Rn) ∼ ‖F‖hΦω(Rn) ∼ ‖f‖hΦ

ω, r(Ω).

Thus, we prove that (i) implies (iv).Now we prove that (iv) implies (i). Let (ρ, q, s)ω be an admissible triplet and

f ∈ D′(Ω) such that

f =∑i

λQ1, iaQ1, i

+∑j

λQ2, jaQ2, j

,

where for each i, aQ1, iis a type (a) local (ρ, q, s)ω-atom supported in the cube

Q1, i, λQ1, i∈ C, and for each j, aQ2, j

is a type (b) local (ρ, q, s)ω-atom supportedin the cube Q2, j and λQ2, j

∈ C.

To finish the proof of this case, we first construct an F ∈ hΦω(R

n) such thatF |Ω = f . For all i and j, let AQ1, i

:= aQ1, i, and if l(Q2, j) ≥ 1, let AQ2, j

:= aQ2, j.

Then both AQ1, iand AQ2, j

are (ρ, q, s)ω-atoms.Now we consider the case when l(Q2, j) < 1. It is known that for all N ∈ N∪{0},

there exist {ϕα}α ⊂ C∞c (B(0, 1)) such that∫

Rn

xβϕα(x) dx = δα, β,

where α, β ∈ Zn+ such that |α|, |β| ≤ N , and δα, β = 1 when α = β, and δα, β = 0

when α �= β; see, for example, [76]. Since Ω� is unbounded, from Lemma 3.15, it

follows that for any cube Q ⊂ Ω such that 2Q ⊂ Ω and 4Q∩ ∂Ω �= ∅, there exist Q

and Q∗ ⊂ Rn such that Q ∪Q∗ ⊂ Q, Q∗ ⊂ (Ω)� and

(4.4) l(Q) = l(Q∗) ∼ l(Q).

Let N := s ≥ �n( qωp−Φ

− 1)�. For all x ∈ Rn and Q2, j satisfying l(Q2, j) < 1, let

AQ2, j(x) := aQ2, j

(x)−∑|α|≤s

bαϕα

(x− xQ∗

2, j

l(Q∗2, j)

),

where for each α ∈ Zn+, bα is a constant which will be determined later and, for

each j, xQ∗2, j

denotes the center of Q∗2, j . For all α ∈ Zn

+ with 0 ≤ |α| ≤ s, to show

that ∫Rn

AQ2, j(x)xα dx = 0,

we set

bα :=1

[l(Q∗2, j)]

|α|+n

∫Rn

aQ2, j(x)

(x− xQ∗

2, j

)α

dx.



Then, for all α ∈ Zn+ with 0 ≤ |α| ≤ s, we see that∫

Rn

AQ2, j(x)xα dx =

∫Rn

aQ2, j(x)xα dx−

∑|β|≤s

bβ

∫Rn

ϕβ

(x− xQ∗

2, j

l(Q∗2, j)

)xα dx

=

∫Rn

aQ2, j(x)xα dx

−∑|β|≤s

bβ [l(Q∗2, j)]

n

∫Rn

ϕβ(x)(l(Q∗

2, j)x+ xQ∗2, j

)α

dx = 0.

Moreover, by (4.4) and Lemma 2.2(iii), we conclude that

‖AQ2, j‖Lq

ω(Rn) ≤ ‖aQ2, j‖Lq

ω(Rn) +∑|α|≤s

|bα|∥∥∥∥∥ϕα

(· − xQ∗

2, j

l(Q∗2, j)

)∥∥∥∥∥Lq

ω(Rn)

� ‖aQ2, j‖Lq

ω(Rn)

+∑|α|≤s

|Q2, j |−n+|α|

n

∣∣∣∣∫Rn

aQ2, j(x)(x− xQ∗

2, j)α dx

∣∣∣∣ [ω(Q∗2, j)]

1q

� [ω(Q2, j)]1q−1[ρ(ω(Q2, j))]

−1

+ |Q2, j |−1‖aQ2, j‖Lq

ω(Rn)

|Q2, j |[ω(Q2, j)]

1q

[ω(Q∗2, j)]

1q

� [ω(Q2, j)]1q−1[ρ(ω(Q2, j))]

−1 ∼ [ω(Q2, j)]1q−1[ρ(ω(Q2, j))]

−1.

Let

F :=∑i

λQ1, iAQ1, i

+∑j

λQ2, jAQ2, j

.

We then see that F ∈ hΦω(R

n), F |Ω = f and

‖F‖hΦω(Rn) � inf

{λ ∈ (0, ∞) :

∑i

ω(Q1, i

)Φ

(|λQ1, i

|λω(Q1, i)ρ(ω(Q1, i))

)

+∑j

ω(Q2, j

)Φ

(|λQ2, j

|λω(Q2, j)ρ(ω(Q2, j))

)≤ 1

⎫⎬⎭ � ‖f‖hp, q, sω (Ω).

From this and the definition of hΦω, r(Ω), we deduce that f ∈ hΦ

ω, r(Ω) and

‖f‖hΦω, r(Ω) � ‖f‖hp, q, s

ω (Ω).

Thus, (vi) implies (i). This finishes the proof of Step I.

Step II. (ii) ⇐⇒ (iii). Obviously, (ii) implies (iii). We now prove that (iii) implies(ii). Let f ∈ D′(Ω) such that f+

Ω, ϕ ∈ LΦω (Ω). Let[

f+Ω, ϕ

]e(x) :=

{f+Ω, ϕ(x), x ∈ Ω,

0, x �∈ Ω.

In what follows, for all g ∈ Lqloc(R

n) with q ∈ (0, ∞) and x ∈ Rn, let

Mq(g)(x) := supt>0

[1

|B(x, t)|

∫B(x, t)

|g(x)|q dy] 1

q

,



and denote M1(g) simply by M(g), which is just the Hardy-Littlewood maximalfunction. Miyachi [65] proved that for all x ∈ Ω,

(4.5) f∗Ω(x) � Mγ

([f+

Ω, ϕ]e)(x),

where γ ∈ (0, 1] such that γ <p−Φ

qω. Similar to the proof of [86, (3.15)], we know

that for any given q ∈ (qω,∞), and all g ∈ L1loc(R

n) and α ∈ (0,∞),

ω ({x ∈ Rn : M(g)(x) > 2α}) � 1

αq

∫{x∈Rn: |g(x)|>α}

|g(x)|qω(x) dx.

From this, it follows that for all α ∈ (0, ∞),

ω({

x ∈ Rn : Mγ

([f+

Ω, ϕ]e)(x) > 2α

})(4.6)

� 1

αγq

∫{x∈Rn: ([f+

Ω, ϕ]e(x))γ>αγ

2 }

([f+

Ω, ϕ]e(x)

)γq

ω(x) dx

∼ σ[f+Ω, ϕ]e

( α

21/γ

)+

1

αγq

∫ ∞

α

21/γ

γqsγq−1σ[f+Ω, ϕ]e(s) ds,

where and in what follows,

σ[f+Ω, ϕ]e(t) := ω({x ∈ Rn : [f+

Ω, ϕ]e(x) > t}).

Choose q ∈ (qω,∞) and p0 ∈ (0, p−Φ) such that γq < p0. Then Φ is of lower type p0and ω ∈ Aq(R

n). From the assumption that Φ is of upper type 1 and of lower typep0, we infer that

Φ(t) ∼∫ t

0

Φ(s)

sds

for all t ∈ (0,∞). By this, (4.5), (4.6), and the upper type 1 and the lower type p0properties of Φ, we conclude that∫

Ω

Φ (f∗Ω(x))ω(x) dx

∼∫Ω

{∫ f∗Ω(x)

0

Φ(t)

tdt

}ω(x) dx

∼∫ ∞

0

Φ(t)

tσf∗

Ω(t) dt �∫ ∞

0

Φ(t)

tσMγ([f+

Ω, ϕ]e)(t) dt

�∫ ∞

0

Φ(t)

t

{σ[f+

Ω, ϕ]e

(t

21/γ

)dt+

1

tγq

∫ ∞

t

21/γ

γqsγq−1σ[f+Ω, ϕ]e(s)

}dt

�∫ ∞

0

Φ(t)

tσ[f+

Ω, ϕ]e(t) dt

+

∫ ∞

0

γqsγq−1σ[f+Ω, ϕ]e(s)Φ(2

1/γs)

{∫ 21/γs

0

(t

21/γs

)p0 1

tγq+1dt

}ds

∼∫ ∞

0

Φ(t)

tσ[f+

Ω, ϕ]e(t) dt ∼∫ ∞

0

Φ(t)

tω({

x ∈ Ω : |f+Ω, ϕ(x)| > t

})dt

∼∫Ω

Φ(f+Ω, ϕ(x))ω(x) dx,



where

σf∗Ω(t) := ω({x ∈ Ω : f∗

Ω(x) > t}).From this and the facts that for all λ ∈ (0,∞),

(f/λ)∗Ω = f∗Ω/λ and (f/λ)+Ω, ϕ = f+

Ω, ϕ/λ,

we deduce that for all λ ∈ (0, ∞),∫Ω

Φ (f∗Ω(x)/λ)ω(x) dx �

∫Ω

Φ(f+Ω, ϕ(x)/λ

)ω(x) dx,

which implies that

‖f∗Ω‖LΦ

ω(Ω) �∥∥∥f+

Ω, ϕ

∥∥∥LΦ

ω(Ω).

This finishes the proof of (ii) =⇒ (iii) and hence Step II.

Step III. (vi) =⇒ (iii). The proof of Step III is similar to that of Proposition 3.4(i).We omit the details.

Step IV. (ii) =⇒ (i). Let f ∈ D′(Ω) such that f+Ω, ϕ ∈ LΦ

ω (Ω). First, we show that

there exists F ∈ hΦω(R

n) such that F |Ω = f and

‖F‖hΦω(Rn) � ‖f+

Ω, ϕ‖LΦω(Ω).

For the strongly Lipschitz domain Ω, let the family of cubes, {Qk}k, be theWhitney decomposition over Ω; namely, there exist a positive constant c1 ∈ (1, 5/4)and a family of cubes, {Qk}k, such that the {Qk}k have disjoint interiors,

⋃k Qk

= Ω,

diam (Qk) ≤ dist (Qk, Ω�) ≤ 4 diam (Qk)

and {c1Qk}k have the bounded intersection property. In the remainder of theproof of this step, for each k, let Q∗

k := 1+c12 Qk, and {ϕk}k be a partition of unity

associated to {Q∗k}k, that is, for all k ∈ N, ϕk ∈ C∞

c (Q∗k), 0 ≤ ϕk ≤ 1 and ϕk ≡ 1

on Qk. For each k, let PQk∈ Ps(R

n) such that for all P ∈ Ps(Rn),

〈fϕk − PQkχQ∗

k, P 〉 = 0,

where Ps(Rn) denotes the linear space of polynomials in n variables of degrees no

more than s. Let

g(x) :=

⎧⎨⎩f(x)−∑k

χQ∗k(x)PQk

(x), x ∈ Ω,

0, x �∈ Ω.

Now, we prove that g ∈ hΦω (R

n) and

‖g‖hΦω(Rn) �

∥∥∥f+Ω, ϕ

∥∥∥LΦ

ω(Ω).

Recall that Miyachi [66, Lemma 1 and (3.25)] proved that for all f ∈ D′(Ω), k ∈ N

and ψ ∈ C∞c (Q∗

k),

|〈f, ψ〉| � |Qk|∑

|α|≤s+1

supy∈Ω

∣∣∂αy (ψ(xQk

+ l(Qk)y))∣∣ infx∈Q∗

k

f∗Ω(x)(4.7)

and

‖PQkχQ∗

k‖L∞(Q∗

k)� inf

x∈Q∗k

f∗Ω(x).(4.8)



Let ψ ∈ C∞c (B(0, 1)) and ∫

Rn

ψ(y) dy = 1.

To prove g ∈ hΦω (R

n), we consider two cases as follows.

Case (i) x ∈ Ω. In this case, from the definition of g, we deduce that

|ψt ∗ g(x)|(4.9)

=

∣∣∣∣∣∑k

∫B(x, t)

[f(y)ϕk(y)− PQk

(y)χQ∗k(y)

]ψt(x− y) dy

∣∣∣∣∣≤

∑{k: |x−xQk

|< c12 l(Qk)}

∫B(x, t)

∣∣[f(y)ϕk(y)− PQk(y)χQ∗

k(y)

]ψt(x− y)

∣∣ dy+

∑{k: |x−xQk

|≥ c12 l(Qk)}

∣∣∣∣∣∫Q∗

k

[f(y)ϕk(y)− PQk

(y)χQ∗k(y)

]

×

⎡⎣ψt(x− y)−∑|α|≤s

∂α(ψt)(x− xQk)(y − xQk

)α

⎤⎦ dy

∣∣∣∣∣∣ =: I1 + I2.

For I1, by (4.8), we know that

I1 �∑

{k: |x−xQk|< c1

2 l(Qk)}

f∗Ω(x)χ{z: |z−xQk

|< c12 l(Qk)}(x)(4.10)

+

{inf

y∈Q∗k

f∗Ω(y)

}χ{z: |z−xQk

|< c12 l(Qk)}(x).

Now we estimate I2. By Taylor’s remainder theorem, we conclude that

I2 ≤∑

{k: |x−xQk|≥ c1

2 l(Qk)}

∣∣∣∣∣∣∫Q∗

k

f(y)ϕk(y)∑

|α|=s+1

∂α(ψt)(x− ξ)(4.11)

× (y − xQk)α dy

∣∣∣∣∣∣+∑

{k: |x−xQk|≥ c1

2 l(Qk)}

∫Q∗

k

∣∣∣∣∣∣χQ∗k(y)PQk

(y)

×∑

|α|=s+1

∂α(ψt)(x− ξ)(y − xQk)α

∣∣∣∣∣∣ dy =: At(x) +Bt(x),

where ξ := θy+ (1− θ)xQkfor some θ ∈ (0, 1). We now estimate At(x) and Bt(x).

From |x− xQk| ≥ c1

2 l(Qk) and t > |x− ξ|, we infer that

t > |x− ξ| ≥ |x− xQk| − θ|y − xQk

| > |x− xQk| − 1 + c1

4l(Qk).

Thus,

t � |x− xQk| � l(Qk).(4.12)

Therefore, from (4.7) and (4.12), we deduce that

At(x) �∑

{k: |x−xQk|≥ c1

2 l(Qk)}

{inf

y∈Q∗k

f∗Ω(y)

}|Qk|

[l(Qk)]s+1

tn+s+1(4.13)



�∑

{k: |x−xQk|≥ c1

2 l(Qk)}

{inf

y∈Q∗k

f∗Ω(y)

}[1 +

|x− xQk|

l(Qk)

]−(n+s+1)

.

For Bt(x), by (4.8) and (4.12), and an estimate similar to At(x), we know that

Bt(x) �∑

{k: |x−xQk|≥ c1

2 l(Qk)}

{inf

y∈Q∗k

f∗Ω(y)

}[1 +

|x− xQk|

l(Qk)

]−(n+s+1)

.(4.14)

Thus, combining (4.9), (4.10), (4.11), (4.13) and (4.14), we conclude that

ψ+(g)(x) := sup0<t≤1

|ψt ∗ g(x)|(4.15)

� f∗Ω(x) +

∑{k: |x−xQk

|≥ c12 l(Qk)}

{inf

y∈Q∗k

f∗Ω(y)

}

×[1 +

|x− xQk|

l(Qk)

]−(n+s+1)

.

Case (ii) x �∈ Ω. In this case, we have

ψt ∗ g(x) =∫B(x, t)∩Ω

∑k

[f(y)ϕk(y)− χQ∗

k(y)PQk

(y)]ψt(x− y) dy

=

∫B(x, t)∩Ω

∑{k: Q∗

k∩B(x, t) �=∅}

[f(y)ϕk(y)− χQ∗

k(y)PQk

(y)]ψt(x− y) dy.

By an argument similar to Case (i), we see that

ψ+(g)(x) �∑

{k: Q∗k∩B(x, t) �=∅}

infy∈Q∗

k

f∗Ω(y)

[1 +

|x− xQk|

l(Qk)

]−(n+s+1)

.(4.16)

Furthermore, by s ≥ �n(qω/p−Φ − 1)�, we know that (n+ s+1)p−Φ > nqω, which,

together with the definitions of qω and p−Φ , implies that there exist p0 ∈ (0, p−Φ) andq ∈ (qω,∞) such that ω ∈ Aq(R

n), Φ is of lower type p0 and (n + s + 1)p0 > nq.From this and Lemma 2.2(iii), we deduce that∫

Rn

[1 +

|x− xQk|

l(Qk)

]−(n+s+1)p0

ω(x) dx � ω(Qk).

By this, (4.15), (4.16), the lower type p0 property of Φ and the equivalence between(ii) and (iii) established in Step II, we conclude that∫

Rn

Φ(ψ+(g)(x)

)ω(x) dx

�∫Ω

Φ (f∗Ω(x))ω(x) dx

+

∫Rn

∑k

Φ

({inf

y∈Q∗k

f∗Ω(y)

}[1 +

|x− xQk|

l(Qk)

]−(n+s+1))ω(x) dx

�∫Ω

Φ (f∗Ω(x))ω(x) dx+

∑k

Φ

(inf

y∈Q∗k

f∗Ω(y)

)



×∫Rn

[1 +

|x− xQk|

l(Qk)

]−(n+s+1)p0

ω(x) dx

�∫Ω

Φ (f∗Ω(x))ω(x) dx+

∑k

Φ

(inf

y∈Q∗k

f∗Ω(y)

)ω(Qk)

�∫Ω

Φ (f∗Ω(x))ω(x) dx �

∫Ω

Φ(f+Ω, ϕ(x)

)ω(x) dx,

which implies that g ∈ hΦω (R

n) and

‖g‖hΦω(Rn) �

∥∥∥f+Ω, ϕ

∥∥∥LΦ

ω(Ω).(4.17)

For all x ∈ Rn, let

R(x) :=∑k

PQk(x)χQ∗

k(x).

Let q ∈ (qω, ∞], s ∈ Z+ and s ≥ �n( qωp−Φ

− 1)�. For all k ∈ N, let

λk :=ω(Q∗

k)ρ(ω(Q∗k))

[ω(Q∗k)]

1q

‖PQk‖Lq

ω(Q∗k)

and

ak :=[ω(Q∗

k)]1q

ω(Q∗k)ρ(ω(Q

∗k))

PQkχQ∗

k

‖PQk‖Lq

ω(Q∗k)

.

Then, obviously,

R =∑k

λkak.

Moreover, ak is a type (b) local (ρ, q, s)ω-atom and, by (4.8) and Lemma 2.2(iii),we further conclude that∑

k

ω(Q∗k)Φ

(|λk|

ω(Q∗k)ρ(ω(Q

∗k))

)

≤∑k

ω(Q∗k)Φ

(‖PQk

‖Lqω(Q∗

k)

[ω(Q∗k)]

1q

)≤∑k

ω(Q∗k)Φ

(‖PQk

‖L∞(Q∗k)

)�∑k

ω(Q∗k)Φ

(inf

y∈Q∗k

f∗Ω(y)

)�∑k

ω(Qk)Φ

(inf

y∈Q∗k

f∗Ω(y)

)�∫Ω

Φ (f∗Ω(y))ω(y) dy.

From this, it follows that

‖R‖hρ, q, sω (Ω) � ‖f∗

Ω‖LΦω(Ω).

By this and the equivalence between (i) and (iv) established in Step I, we see that

there exists R ∈ hΦω (R

n) such that R|Ω = R and∥∥∥R∥∥∥hΦω(Rn)

� ‖R‖hΦω, r(Ω) ∼ ‖R‖hρ, q, s

ω (Ω) � ‖f∗Ω‖LΦ

ω(Ω) �∥∥∥f+

Ω, ϕ

∥∥∥LΦ

ω(Ω).(4.18)

Let F := g + R. By (4.17) and (4.18), we have F ∈ hΦω (R

n), F |Ω = f and

‖F‖hΦω(Rn) � ‖g‖hΦ

ω(Rn) +∥∥∥R∥∥∥

hΦω(Rn)

�∥∥∥f+

Ω, ϕ

∥∥∥LΦ

ω(Ω),

which completes the proof of Step VI and hence Theorem 1.7. �




In this section, we give the proof of Theorem 1.8. In what follows, we alwaysassume that Ω is a bounded, simply connected, semiconvex domain in Rn, and GD

the Dirichlet Green operator for the problem (1.1). Denote the integral kernel ofGD by GD. We first recall the notion of semiconvex domains in Rn and some usefulestimates for GD; see [26].

Definition 5.1. (i) Let O be an open set in Rn. The collection of semiconvexfunctions on O consists of continuous functions u : O �→ R with the propertythat there exists a positive constant C such that for all x, h ∈ Rn with the ballB(x, |h|) ⊂ O,

2u(x)− u(x+ h)− u(x− h) ≤ C|h|2.The best constant C above is referred to as the semiconvexity constant of u.

(ii) A nonempty, proper open subset Ω of Rn is called semiconvex providedthere exist b, c ∈ (0,∞) with the property that for every x0 ∈ ∂Ω, there exist an(n − 1)-dimensional affine variety H ⊂ Rn passing through x0, a choice N of theunit normal to H, and an open set

C := {x+ tN : x ∈ H, |x− x0| < b, |t| < c},

called a coordinate cylinder near x0 (with axis along N), such that for some semi-convex function ϕ : H → R satisfying

C ∩ Ω = C ∩ {x+ tN : x ∈ H, t > ϕ(x)},

C ∩ ∂Ω = C ∩ {x+ tN : x ∈ H, t = ϕ(x)},

C ∩ Ω�= C ∩ {x+ tN : x ∈ H, t < ϕ(x)},

we have

ϕ(x0) = 0 and |ϕ(x)| < c/2 if |x− x0| ≤ b.

The following Lemma 5.2 was established in [36, 31]. Recall that for all y ∈ Ω,

δ(y) := dist (y, ∂Ω).

Lemma 5.2. Let Ω and GD be as in Theorem 1.8. Denote the integral kernel ofGD by GD. Then there exists a positive constant C such that for all x, y ∈ Ω withx �= y,

(i) 0 ≤ GD(x, y) ≤ C|x− y|2−n;(ii) |∇xGD(x, y)| ≤ C|x− y|1−n;(iii) |∇xGD(x, y)| ≤ Cδ(y)|x− y|−n;(iv) |∇yGD(x, y)| ≤ C|x− y|1−n;(v) |∇x∇yGD(x, y)| ≤ C|x− y|−n.

The following Lemma 5.3 is just [26, Theorem 4.1].

Lemma 5.3. Let Ω and GD be as in Theorem 1.8. Then the operators in (1.3),originally defined on C∞(Ω), can be extended to bounded operators on Lp(Ω) forp ∈ (1, 2].

Now we prove Theorem 1.8 by using Lemmas 5.2 and 5.3 and Theorem 1.7.



Proof of Theorem 1.8. We first prove Theorem 1.8(i). Fix i, j ∈ {1, · · · , n} and

denote by T the operator ∂2GD

∂xi∂xj. First let f ∈ hΦ

ω, r(Ω)∩L2(Ω). By the assumption

that rω > 22−qω

, we know that rωqωrω−1 < 2. Take q ∈ ( rωqω

rω−1 , 2] and q1 ∈ ( rωrω−1 ,

2qω]

such that qq1

> qω. Then

2

2− qω≤ q′1 < rω,

where and in what follows, 1q1

+ 1q′1

= 1. Thus, ω ∈ RHq′1(Rn) and ω ∈ Aq/q1(R

n).

By Theorem 1.7, we conclude that

(5.1) f =∑

type (a)-atoms

λ1, ka1, k +∑

type (b)-atoms

λ2,ma2,m,

where {a1, k}k and {a2,m}m are respectively sequences of type (a) local (ρ, q, 0)ω-atoms and type (b) local (ρ, q, 0)ω-atoms, and ({λ1, k}k∪{λ2,m}m) ⊂ C. Moreover,

(5.2) Λ ({λ1, ka1, k}k ∪ {λ2,ma2,m}m) ∼ ‖f‖hΦω, r(Ω).

To finish the proof of Theorem 1.8(i), we only need to show that for any type(a) local (ρ, q, 0)ω-atom or any type (b) local (ρ, q, 0)ω-atom a supported in thecube Q0 and any λ ∈ C,

(5.3)

∫Ω

Φ(T (λa)(x))ω(x) dx � ω(Q0)Φ

(|λ|

ω(Q0)ρ(ω(Q0))

).

Indeed, if (5.3) holds, then by (5.1) and the assumption that Φ is subadditive, weknow that for all λ ∈ (0,∞),∫

Ω

Φ

(T

(f

λ

)(x)

)ω(x) dx

≤∑k

∫Ω

Φ

(T

(λ1, ka1, k

λ

)(x)

)ω(x) dx

+∑m

∫Ω

Φ

(T

(λ2,ma2,m

λ

)(x)

)ω(x) dx

�∑k

ω(Q1, k)Φ

(|λ1, k|


)+∑m

ω(Q2, m)Φ

(|λ2,m|

λω(Q2, m)ρ(ω(Q2,m))

),

where for each k and m, supp (a1, k) ⊂ Q1, k and supp (a2,m) ⊂ Q2,m, which,together with Theorem 1.7 and (5.2), implies that

‖T (f)‖LΦω(Ω) � ‖f‖hΦ

ω, r(Ω).

From this, the fact that hΦω, r(Ω)∩L2(Ω) is dense in hΦ

ω, r(Ω) and a density argument,we deduce that Theorem 1.8(i) holds.

Now we prove (5.3) by considering the following three cases for Q0.

Case i) l(Q0) ≥ 1. In this case, by

ω ∈ RHq′1(Rn) ∩ Aq/q1(R

n),



Jensen’s inequality, Holder’s inequality, Lemma 5.3, (iii) and (v) of Lemma 2.2, theassumptions that Ω is bounded and l(Q0) ≥ 1, we conclude that∫

Ω

Φ(T (λa)(x))ω(x) dx(5.4)

≤ ω(Ω)Φ

(1

ω(Ω)

{∫Ω

|T (λa)(x)|q1 dx} 1

q1{∫

Ω

[ω(x)]q′1 dx

} 1q′1

)

� ω(Ω)Φ

(1

|Ω|1q1

‖λa‖Lq1 (Q0)

)� ω(Ω)Φ

(|λ||Q0|

1q1

|Ω|1q1 [ω(Q0)]

1q

‖a‖Lqω(Q0)

)

� ω(Q0)Φ

(|λ|

ω(Q0)ρ(ω(Q0))

).

Case ii) l(Q0) < 1, 2Q0 ⊂ Ω and 4Q0 ∩ ∂Ω �= ∅. In this case, we have∫Ω

Φ(T (λa)(x))ω(x) dx =

∫4Q0∩Ω

Φ(T (λa)(x))ω(x) dx(5.5)

+∞∑j=2

∫(2j+1Q0\2jQ0)∩Ω

· · · =: I1 + I2.

Similar to the proof of (5.4), we have

I1 � ω(Q0)Φ

(|λ|

ω(Q0)ρ(ω(Q0))

).(5.6)

Now we deal with I2. For all j ∈ N with j ≥ 2, let

Rj(Q0) := (2j+1Q0 \ 2jQ0) ∩ Ω.

Similar to the proof of [26, (5.34)] (or [83, p. 346, (8)]), we know that∫Rj(Q0)

|T (λa)(x)|ω(x) dx � |λ|l(Q0)

[2j l(Q0)]n+1‖a‖L1(Q0)ω(Rj(Q0))(5.7)

� 2−j(n+1)|λ|ω(Rj(Q0))

[ω(Q0)]1q

‖a‖Lqω(Q0)

� 2−j(n+1)|λ| ω(Rj(Q0))

ω(Q0)ρ(ω(Q0)).

By the assumption that nqω < (n + 1)p−Φ , we know that there exist q0 ∈ (qω,∞)

and p0 ∈ (0, p−Φ) such that Φ is of lower type p0 and nq0 < (n + 1)p0. Moreover,by the definition of qω, we see that ω ∈ Aq0(R

n). Then, from Jensen’s inequality,Holder’s inequality, (iii) and (v) of Lemma 2.2, the lower type p0 property of Φ,(5.7) and nq0 < (n+ 1)p0, it follows that

I2 ≤∞∑j=2

ω(Rj(Q0))Φ

(1

ω(Rj(Q0))

∫Rj(Q0)

|T (λa)(x)|ω(x) dx)

(5.8)

�∞∑j=2

ω(Rj(Q0))Φ

(2−k(n+1) |λ|

ω(Q0)ρ(ω(Q0))

)

�∞∑j=2

2−j[(n+1)p0−nq0]ω(Q0)Φ

(|λ|

ω(Q0)ρ(ω(Q0))

)



� ω(Q0)Φ

(|λ|

ω(Q0)ρ(ω(Q0))

).

Thus, by (5.5), (5.6) and (5.8), we conclude that (5.3) holds in this case.

Case iii) l(Q0) < 1 and 4Q0 ⊂ Ω. In this case, we have∫Ω

Φ(T (λa)(x))ω(x) dx(5.9)

=

∫2Q0

Φ(T (λa)(x))ω(x) dx+∞∑j=1

∫(2j+1Q0\2jQ0)∩Ω

· · · =: J0 +∞∑j=1

Jj .

For j ∈ {0, 1, 2}, similar to the estimate of (5.6), we know that

(5.10) Jj � ω(Q0)Φ

(|λ|

ω(Q0)ρ(ω(Q0))

).

Now we deal with Jj for j ∈ N with j ≥ 3. Pick ϕ ∈ C∞c (R+) satisfying ϕ(t) ≡ 0 if

t ≤ 12 or t ≥ 4, and ϕ(t) ≡ 1 if 1 ≤ t ≤ 2. Furthermore, for each j ≥ 3, let

ϕj(x) := ϕ(|x− xQ0|/(2jl(Q0)))

for all x ∈ Rn, where xQ0denotes the center of Q0. Similar to the proof of [26,

p. 49, (5.29)], we have∫Rj(Q0)

|T (λa)(x)|ω(x) dx �∫Rj(Q0)

|GD(λa)(x)Δϕj(x)|ω(x) dx(5.11)

+

∫Rj(Q0)

|∇xGD(λa)(x) · ∇ϕj(x)|ω(x) dx.

In this case, we know that a is a type local (a) (ρ, q, 0)ω-atom, and hence∫Ω

a(x) dx = 0.

From this, Lemma 5.2(iv) and Holder’s inequality, it follows that for all x ∈ Rj(Q0),

|GD(λa)(x)| =∣∣∣∣λ ∫

Q0

GD(x, y)a(y) dy

∣∣∣∣=

∣∣∣∣λ ∫Q0

[GD(x, y)−GD(x, xQ0)]a(y) dy

∣∣∣∣≤ |λ|

∫Q0

|∇yGD(x, y1)||y − xQ0||a(y)| dy

� |λ|∫Q0

|y − xQ0|

|x− y1|n−1|a(y)| dy � |λ|l(Q0)

|x− xQ0|n−1

‖a‖L1(Rn)

� |λ|l(Q0)

|x− xQ0|n−1

|Q0|[ω(Q0)]

1q

‖a‖Lqω(Q0),

where y1 ∈ Q0 depends on y and xQ0, and in the penultimate inequality we used

the fact that |x− y1| ∼ |x−xQ0|, which, together with the fact that for all x ∈ Rn,

|Δϕj(x)| � [2j l(Q0)]−2, implies that

(5.12)

∫Ω

|GD(λa)(x)Δϕj(x)|ω(x) dx � 2−j(n+1) |λ|ω(Rj(Q0))

ω(Q0)ρ(ω(Q0)).



Moreover, by the fact that∫Q0

a(x) dx = 0, Lemma 5.2(v) and Holder’s inequality,

we know that for all x ∈ Rj(Q0),

|∇xGD(λa)(x)| =∣∣∣∣λ ∫

Q0

∇xGD(x, y)a(y) dy

∣∣∣∣=

∣∣∣∣λ ∫Q0

[∇xGD(x, y)−∇xGD(x, xQ0)]a(y) dy

∣∣∣∣≤ |λ|

∫Q0

|y − xQ0| supy1∈Q0

|∇x∇yGD(x, y1)||a(y)| dy

� |λ|l(Q0)

|x− xQ0|n ‖a‖L1(Q0) �

|λ|[l(Q0)]n+1‖a‖Lq

ω(Q0)

|x− xQ0|n[ω(Q0)]

1q

,

which, together with the fact that for all x ∈ Rn, |∇ϕj(x)| � [2j l(Q0)]−1, implies

that

(5.13)

∫Ω

|∇GD(λa)(x) · ∇ϕj(x)|ω(x) dx � 2−j(n+1) |λ|ω(Rj(Q0))

ω(Q0)ρ(ω(Q0)).

Thus, by (5.11), (5.12) and (5.13), we conclude that for each j ∈ N with j ≥ 3,∫Rj(Q0)

|T (λa)(x)|ω(x) dx � 2−j(n+1) |λ|ω(Rj(Q0))

ω(Q0)ρ(ω(Q0)),

which, together with Jensen’s inequality, (iii) and (v) of Lemma 2.2, the lower typep0 property of Φ and nq0 < (n+ 1)p0, implies that

∞∑j=3

Jj ≤∞∑j=3

ω(Rj(Q0))Φ

(1

ω(Rj(Q0))

∫Rj(Q0)

|T (λa)(x)|ω(x) dx)

�∞∑j=3

ω(Rj(Q0))Φ

(2−j(n+1) |λ|

ω(Q0)ρ(ω(Q0))

)

�∞∑j=3

2−j[(n+1)p0−nq0]ω(Q0)Φ

(|λ|

ω(Q0)ρ(ω(Q0))

)

� ω(Q0)Φ

(|λ|

ω(Q0)ρ(ω(Q0))

).

This, combined with (5.9) and (5.10), implies that (5.3) holds in this case and hencecompletes the proof of Theorem 1.8(i).

Now we prove Theorem 1.8(ii). We borrow some ideas from [26] and [83]. Firstlet

f ∈ hΦω, r(Ω) ∩ L2(Ω).

By the assumption that rω > 22−qω

and the definition of qω, we know that there

exists q2 ∈ (qω, 2) such that

2

2− q2< rω.

Then ( 2q2)′ < rω and hence ω ∈ A(2/q2)′(R

n). Take q3 ∈ (2qω,∞). Then ω ∈Aq3/2(R

n). Then by Theorem 1.7, we know that there exist a sequence {a1, k}k of



type (a) local (ρ, q2, 0)ω-atoms, a sequence {a2,m}m of type (b) local (ρ, q3, 0)ω-atoms and ({λ1, k}k ∪ {λ2,m}m) ⊂ C such that (5.1) and (5.2) hold. Moreover, byLemma 5.3, we see that

T (f) =∑

type (a) atoms

λ1, kT (a1, k) +∑

type (b) atoms

λ2,mT (a2,m) in L2(Ω).

To finish the proof of Theorem 1.8(ii), we only need to show that for all type (a)local (ρ, q2, 0)ω-atoms or type (b) local (ρ, q2, 0)ω-atoms a supported in the cubeQ0 and all λ ∈ C\{0}, there exist a sequence {bs}s of type (a) local (ρ, q1, 0)-atomsand type (b) local (ρ, q1, 0)-atoms, and {μs}s ⊂ C such that

(5.14) T (λa) =∑s

μsbs

and for all μ ∈ (0,∞),

(5.15)∑s

ω(Qs)Φ

(|μs|

μω(Qs)ρ(ω(Qs))

)� ω(Q0)Φ

(|λ|

μω(Q0)ρ(ω(Q0))

),

where for each s, supp (bs) ⊂ Qs.Indeed, if (5.14) and (5.15) hold, then for each k and m, there exist sequences

{b1, k, s}k, s and {b2, m, s}m, s of type (a) local (ρ, q1, 0)-atoms and type (b) local(ρ, q1, 0)-atoms, and {μ1, k, s}k, s ∪ {μ2, m, s}m, s ⊂ C such that

T (f) =∑k, s

λ1, kμ1, k, sb1, k, s +∑m, s

λ2,mμ2, m, sb2, m, s,

and for all λ ∈ (0,∞),∑k, s

ω(Q1, k, s)Φ

(|λ1, kμ1, k, s|

λω(Q1, k, s)ρ(ω(Q1, k, s))

)

+∑m, s

ω(Q2, m, s)Φ

(|λ2,mμ2,m, s|

λω(Q2,m, s)ρ(ω(Q2,m, s))

)

�∑k

ω(Q1, k)Φ

(|λ1, k|


)+∑m

ω(Q2,m)Φ

(|λ2,m|

λω(Q2,m)ρ(ω(Q2,m))

),

where for each k, m, s, supp (a1, k) ⊂ Q1, k, supp (a2,m) ⊂ Q2, m, supp (b1, k, s) ⊂Q1, k, s and supp (b2,m, s) ⊂ Q2, m, s, which, together with Theorem 1.7 and (5.2),implies that T (f) ∈ hΦ

ω, r(Ω) and

‖T (f)‖hΦω, r(Ω) � ‖f‖hΦ

ω, r(Ω).

From this, the fact that hΦω, r(Ω)∩L2(Ω) is dense in hΦ

ω, r(Ω) and a density argument,we deduce Theorem 1.8(ii).

Now we prove (5.14) and (5.15) by considering the following two cases for a.

Case i) a is a type (a) local (ρ, q3, 0)ω-atom. Recall that the standard fundamentalsolution of the Laplace operator

Δ :=

n∑i=1

∂2

∂x2i



on Rn (n ≥ 2) is given by

(5.16) Γ(x) :=

⎧⎪⎨⎪⎩1

2πln |x|, when n = 2,

cn|x|n−2

, when n ≥ 3,

where cn := [(n− 2)ωn]−1 and ωn denotes the area of the unit sphere in Rn. This

allows us to solve the Poisson problem for the Laplacian in the whole space viaintegral operators. Indeed, as is well known, the Newtonian potential

E(f)(x) :=

∫Ω

Γ(x− y)f(y) dy, x ∈ Ω,

satisfies ΔE(f) = f in Ω when f ∈ C∞(Ω).For each y ∈ Ω, let U(·, y) be the solution of the Dirichlet problem

(5.17)

{ΔU(·, y) = 0, in Ω,

U(x, y) = Γ(x− y), for x ∈ ∂Ω.

Then the Green function GD for ΔD on Ω (which is the integral kernel of theDirichlet Green potential GD) can be expressed as

GD(x, y) = Γ(x− y)− U(x, y), x, y ∈ Ω, x �= y.

As a consequence, the solution of the inhomogeneous Dirichlet problem (1.1) isgiven by

GD(f)(x) =

∫Ω

GD(x, y)f(y) dy(5.18)

=

∫Ω

Γ(x− y)f(y) dy −∫Ω

U(x, y)f(y) dy

=: E(f)(x)− U(f)(x),

where f ∈ C∞(Ω) and x ∈ Ω.By abuse of notation, we denote by a the zero extension of a out of Ω. Then

similar to the proof of [86, Theorem 8.2], we know that

∂2E(λa)

∂xi∂xj∈ hΦ

ω (Rn) and

∥∥∥∥∂2E(λa)

∂xi∂xj

∥∥∥∥hΦω(Rn)

� ‖λa‖hΦω(Rn),

which, together with the definition of hΦω, r(Ω), implies that

(5.19)

∥∥∥∥∂2E(λa)

∂xi∂xj

∥∥∥∥hΦω, r(Ω)

≤∥∥∥∥∂2E(λa)

∂xi∂xj

∥∥∥∥hΦω(Rn)

� ‖λa‖hΦω(Rn).

For i, j ∈ {1, · · · , n}, let

Hi, j(λa) :=∂2U(λa)

∂xi∂xj.

From (5.17) and (5.18), we infer that Hi, j(λa) is a harmonic function in Ω. Thus,an application of Theorem 1.7, in which we take the function ϕ to be radial, yields,on account of the mean value property for harmonic functions, that for all x ∈ Ω,

(Hi, j(λa))+Ω, ϕ (x) = sup

0<t<δ(x)/c0

∣∣∣∣∫Ω

Hi, j(λa)(y)ϕt(x− y) dy

∣∣∣∣(5.20)

= |Hi, j(λa)(x)| .



Similar to the proof of Theorem 1.8(i), we know that∥∥∥∥∂2E(λa)

∂xi∂xj

∥∥∥∥LΦ

ω(Rn)

� ‖λa‖hΦω(Rn).

By this, (5.20), Theorem 1.7 and Theorem 1.8(i), we conclude that

‖Hi, j(λa)‖hΦω, r(Ω) =

∥∥∥(Hi, j(λa))+Ω, ϕ

∥∥∥LΦ

ω(Ω)= ‖Hi, j(λa)‖LΦ

ω(Ω)

�∥∥∥∥∂2E(λa)

∂ixi∂jxj

∥∥∥∥LΦ

ω(Ω)

+

∥∥∥∥∂2GD(λa)

∂xi∂xj

∥∥∥∥LΦ

ω(Ω)

� ‖λa‖hΦω(Rn),

which, together with (5.19), implies that T (λa) ∈ hΦω, r(Ω) and

‖T (λa)‖hΦω, r(Ω) � ‖λa‖hΦ

ω(Rn).

From this and Theorem 1.7, we infer that (5.14) and (5.15) hold in this case.

Case ii) a is a type (b) local (ρ, q3, 0)ω-atom. Let

Rk(Q0) :=(2k+1Q0 \ 2kQ0

)∩ Ω

when k ≥ 3, R0(Q0) := 8Q0 ∩ Ω and

JΩ := {k ∈ N : k ≥ 3, |Rk(Q0)| > 0} .For all k ∈ N ∪ {0}, let χk := χRkQ0

, χk := χk

|Rk(Q0)| and

mk :=

∫Rk(Q0)

T (λa) dx.

Then, we have

T (λa) = T (λa)χ0 +∑k∈JΩ

T (λa)χk.

Let Q0 ⊂ Rn satisfy x˜Q0

∈ Ω�, l(Q0) = l(Q0) and dist (Q0, Q0) ∼ l(Q0), which

implies that there exists Q∗0 ⊂ Rn such that (8Q0 ∪ Q0) ⊂ Q∗

0 and l(Q∗0) ∼ l(Q0).

Moreover, let

H0 := T (λa)χ0 −1

|Q0 ∩ Ω|

{∫R0(Q0)

T (λa)(y) dy

}χ

˜Q0∩Ω� .

We have ∫Rn

H0(x) dx = 0 and supp (H0) ⊂ Q∗0.

From ω ∈ RH(2/q2)′(Rn) ∩ Aq3/2(R

n), Holder’s inequality and Lemma 5.3, itfollows that

‖H0‖Lq2ω (Rn) � ‖T (λa)‖Lq2

ω (R0(Q0))

≤{∫

R0(Q0)

|T (λa)(x)|2 dx

} 12

⎧⎨⎩(∫

R0(Q0)

[ω(x)](2q2

)′ dx

) 1

( 2q2

)′⎫⎬⎭

1q2

≤ ‖λa‖L2(Ω)

{ω(R0(Q0)

|Q0|q2/2

} 1q2

≤ ‖λa‖Lq3ω (Ω)

|Q0|12

[ω(Q0)]1q3

[ω(R0(Q0))]1q2

|Q0|12

� |λ|[ω(Q0)]1q2

−1[ρ(ω(Q0))]−1 ∼ |λ|[ω(Q∗

0)]1q2

−1[ρ(ω(Q∗0))]

−1.



Thus, there exists a positive constant C such that H0

˜C|λ| is a (ρ, q1, 0)ω-atom.

If k ∈ JΩ, by Fact 1 in the proof of Proposition 3.13, we know that there exist{Qk, j}j ⊂ Rn such that for all j, 2Qk, j ⊂ Ω, 4Qk, j ∩ ∂Ω �= ∅ and

Rk(Q0) =⋃j

Qk, j .

From Lemma 3.15, it follows that for each k and j, there exists Qk, j ⊂ Ω� such

that l(Qk, j) = l(Qk, j) and dist (Qk, j , Qk, j) ∼ l(Qk, j). For each k and j, takecube Q∗

k, j ⊂ Rn such that

Qk, j ∪ Qk, j ⊂ Q∗k, j and l(Q∗

k, j) ∼ l(Qk, j).

By this, Holder’s inequality, Lemma 5.3, (iii) and (v) of Lemma 2.2, we see that

‖Hk, i‖Lq2ω (Rn) � ‖T (λa)‖Lq2

ω (Qk, j)

+[ω(Qk, j)]

1/q2

|Qk, j |‖T (λa)‖Lq2

ω (Qk, j)

(∫˜Qk, j

[ω(x)]−q2

′q2 dx

)1/q2′

≤ C ‖T (λa)‖Lq2ω (Qk, j)

,

where C is a positive constant. Let

λk, i := C‖T (λa)‖Lq2ω (Qk, j)

[ω(Q∗k, j)]

1−1/q2ρ(ω(Q∗k, j))

and bk, i := Hk, i/λk, i. We then have Hk, i = λk, jbk, j and bk, i is a (ρ, q2, 0)-atom.Moreover, by the assumption that nqω < (n + 1)p−Φ and the definitions of qω and

p−Φ , we know that there exist q ∈ (qω,∞) and p0 ∈ (0, p−Φ) such that Φ is of lowertype p0 and nq < (n+ 1)p0. Similar to the proof of (5.7), we know that∫

Rk(Q0)

|T (λa)(x)|q2 ω(x) dx � 2−k(n+1)q2‖λa‖q2

Lq2ω (Q0)

ω(Rk(Q0))

ω(Q0).

This, combined with Holder’s inequality, the lower type p0 and the upper type 1properties of Φ, Lemma 2.2(iii) and nq < (n+ 1)p0, implies that∑

k

∑i

ω(Q∗k, i)Φ

(|λ| ‖T (a)‖Lq2

ω (Qk, i)[ω(Q∗

k, i)]1−1/q2ρ(ω(Q∗

k, i))

[ω(Q∗k, i)]ρ(ω(Q

∗k, i))

)

�∑k

∑i

ω(Q∗k, i)

[ω(Rk(Q0))]1/q2

[ω(Qk, i)]1/q2

[‖T (a)‖Lq2

ω (Qk, i)

‖T (a)‖Lq2ω (Rk(Q0))

]p0

× Φ

( |λ|‖T (a)‖Lq2ω (Rk(Q0))

[ω(Rk(Q0))]1/q2

)

�∑k

[∑i

ω(Q∗k, i)

]1/q2′

[ω(Rk(Q0))]1/q2

‖T (a)‖p0

Lq2ω (Rk(Q0))

[∑i

∫Qk, i

|T (a)(x)|q2ω(x) dx]p0/q2

× Φ

(2−k(n+1)|λ|

ω(Q0)ρ(ω(Q0))

)

�∑k

[∫Rk(Q0)

ω(x) dx

]1/q2′

[ω(Rk(Q0))]1/q2

‖T (a)‖p0

Lq2ω (Rk(Q0))



× ‖T (a)‖p0

Lq2ω (Rk(Q0))

Φ

(2−k(n+1)|λ|

ω(Q0)ρ(ω(Q0))

)�∑k

ω(Rk(Q0))Φ

(2−k(n+1)|λ|

ω(Q0)ρ(ω(Q0))

)�∑k

2−k[(n+1)p0−nq]ω(Q0)Φ

(|λ|

ω(Q0)ρ(ω(Q0))

)� ω(Q0)Φ

(|λ|

ω(Q0)ρ(ω(Q0))

).

Therefore, we obtain T (λa) ∈ hΦω, r(Ω) and

‖T (λa)‖hΦω, r(Ω) � inf

{s ∈ (0, ∞) : ω(Q∗

0)Φ

(|λ|

sω(Q∗0)ρ(ω(Q

∗0))

)

+∑k, i

ω(Q∗k, i)Φ

(λk, i

sω(Q∗k, i)ρ(ω(Q

∗k, i))

)≤ 1

⎫⎬⎭� inf

{s ∈ (0, ∞) : ω(Q0)Φ

(|λ|

sω(Q0)ρ(ω(Q0))

)≤ 1

}.

From this and Theorem 1.7, we deduce that (5.14) and (5.15) hold in this case.This finishes the proof of Theorem 1.8(ii) and hence Theorem 1.8. �


In this section, we give the proof of Theorem 1.9. In what follows, we alwaysassume that Ω is a bounded convex domain in Rn. We first establish some auxiliarylemmas.

Let L := −Δ with the Neumann boundary condition on Ω. Denote by D(L)the domain of the operator L, and by Lk the k-fold composition of L with itself,in the sense of unbounded operators. In what follows, to simplify the notation, wejust use BΩ for B(xB, rB) ∩ Ω. For given λ ∈ (0,∞), we denote by λBΩ the setB(xB, λrB) ∩ Ω. Let

UΩ0 (B) := BΩ and UΩ

j (B) := (2jBΩ \ 2j−1BΩ) for j ∈ N.

Definition 6.1. Let Φ satisfy Assumption (A), ω ∈ A∞(Rn) and Ω be an opensubset of Rn. Let p−Φ , ρ and qω be respectively as in (2.12), (2.14) and (2.5). Let

(6.1) M ∈ N and M >

⌊n

2

(qω

p−Φ− 1

)⌋.

A measurable function a on Ω is called a local (ρ, 2, M)ω-atom if there exists aball B of Rn centered in Ω (but not necessarily included in Ω) with radius rB ≤2 diam (Ω) such that

‖a‖L2(Ω) ≤ |BΩ| 12[ω(BΩ

)ρ(ω(BΩ

))]−1

and either(i) rB > 1, or



(ii) rB ≤ 1 and there exists a function b ∈ D(LM ) such that a = LM b,supp (Lkb) ⊂ B ∩ Ω for all k ∈ {0, 1, · · · , M}, and∥∥(r2BL)kb∥∥L2(Ω)

≤ r2MB∣∣BΩ

∣∣ 12 [ω (BΩ)ρ(ω(BΩ

))]−1

for all k ∈ {0, 1, · · · , M}.

Definition 6.2. Let Φ satisfy Assumption (A), ω ∈ A∞(Rn), Ω be a bounded,simply connected convex domain of Rn and L = −Δ with the Neumann boundarycondition. Let ρ and M be respectively as in (2.14) and (6.1). A function f ∈ L2(Ω)

is said to be in hΦL, at,M, ω(Ω) if there exist a sequence {ai}i of (ρ, 2, M)ω-atoms

and {λi}i ⊂ C such that f =∑

i λiai in L2(Ω) and∑i

ω(BΩ

i

)Φ

(|λi|

ω(BΩi )ρ(ω(B

Ωi ))

)< ∞,

where for each i, supp ai ⊂ BΩi ∩ Ω. Moreover, letting

Λ({λiai}i) := inf

{λ ∈ (0,∞) :

∑i

ω(BΩ

i

)Φ

(|λi|

λω(BΩi )ρ(ω(B

Ωi ))

)≤ 1

},

the quasi-norm of f ∈ hΦL, at,M, ω(Ω) is defined by

‖f‖hΦL, at,M, ω(Ω) := inf {Λ({λiai}i)} ,

where the infimum is taken over all the decompositions of f as above. The weightedlocal atomic Orlicz-Hardy space hΦ

L, at,M, ω(Ω) is defined to be the completion of

hΦL, at,M, ω(Ω) in the quasi-norm ‖ · ‖hΦ

L, at, M, ω(Ω).

Lemma 6.3. Let Φ, L, Ω and M be as in Definition 6.2 and ω as in Theorem 1.4.Then the spaces hΦ

ω, z(Ω) and hΦL, at,M, ω(Ω) coincide with equivalent quasi-norms.

Proof. Similar to the proof of [26, Theorem 3.5], we have

hΦNh, ω

(Ω) ∩ L2(Ω) = hΦL, at,M, ω(Ω) ∩ L2(Ω)

with equivalent quasi-norms. From this, the following two facts that

hΦNh, ω

(Ω) ∩ L2(Ω) and hΦL, at,M, ω(Ω) ∩ L2(Ω)

are, respectively, dense in hΦNh, ω

(Ω) and hΦL, at,M, ω(Ω), and a density argument, we

deduce that the spaces hΦNh, ω

(Ω) and hΦL, at,M, ω(Ω) coincide with equivalent quasi-

norms. By the assumption that Ω is a bounded convex domain of Rn and [26,Lemma 2.8], we know that (G1) holds with μ = 1 for L. From this and Theorem1.4(iii), we infer that the spaces hΦ

ω (Ω) and hΦL, at,M, ω(Ω) coincide with equivalent

quasi-norms. This finishes the proof of Lemma 6.3. �

To show Theorem 1.9, we need the following useful estimates.

Lemma 6.4. Let Ω and L be as in Definition 6.2. Denote by {Kt}t≥0 the kernelsof the semigroup {e−tL}t≥0. Let q ∈ [1, 2). Then there exist positive constants γand C such that for all y ∈ Ω and s, t ∈ (0,∞),

(6.2)

[∫{x∈Ω: |x−y|≥

√s}|∇2

xKt(x, y)|q dx] 1

q

≤ Ct−1|BΩ(y,√t)| 1q−1e−γ s

t .



Furthermore, for each k ∈ N, there exist positive constants γk and C(k), depending

on k, such that the k-th order time derivative dk

dtkKt of the kernel Kt satisfies that

for all y ∈ Ω and s, t ∈ (0,∞),[∫{x∈Ω: |x−y|≥

√s}

∣∣∣∣∇2x

(dk

dtkKt(x, y)

)∣∣∣∣q dx

] 1q

(6.3)

≤ C(k)t−(k+1)|BΩ(y,√t)| 1q−1e−γk

st .

Proof. We first prove (6.2). It was shown in [26, Proposition 4.15] that for someγ1 ∈ (0,∞), there exists a positive constant C(γ1) such that for all y ∈ Ω andt ∈ (0,∞),

(6.4)

∫Ω

∣∣∇2xKt(x, y)

∣∣2 e γ1|x−y|2t dx ≤ C(γ1)t

−2|B(y,√t)|−1.

Moreover, it was obtained in [26, Lemma 4.13] that for any γ2 ∈ (0,∞), there existsa positive constant C(γ2) such that for all s ∈ [0,∞), t ∈ (0,∞) and y ∈ Ω,∫

{x∈Ω: |x−y|≥√s}

e−2γ2|x−y|2

t dx ≤ C(γ2)|BΩ(y,√t)|e−γ2

st .

This, together with Holder’s inequality and (6.4), implies that[∫{x∈Ω: |x−y|≥

√s}|∇2

xKt(x, y)|q dx] 1

q

≤[∫

Ω

∣∣∇2xKt(x, y)

∣∣2 e γ1|x−y|2t dx

] 12

[∫{x∈Ω: |x−y|≥

√s}e−

γ1q2−q

|x−y|2t dx

] 1q−

12

� t−1|BΩ(y,√t)|− 1

2 e−γs/t|BΩ(y,√t)| 1q− 1

2 � t−1e−γs/t|BΩ(y,√t)| 1q−1,

where γ := γ1/4, which implies that (6.2) holds.The proof of (6.3) is similar to that of [26, Lemma 4.12]. We omit the details.

This finishes the proof of Lemma 6.4. �

The following Lemma 6.5 is just [26, Theorem 4.2].

Lemma 6.5. Let Ω be a bounded, simply connected semiconvex domain in Rn andGN the Neumann Green operator for the problem (1.2). Then the operators in(1.4), originally defined on C∞(Ω), can be extended to bounded operators on Lp(Ω)for p ∈ (1, 2].

Now we prove Theorem 1.9 by using Lemmas 6.3 and 6.4.

Proof of Theorem 1.9. We borrow some ideas from [26]. Fix m, s ∈ {1, · · · , n}and denote by T the operator ∂2GN

∂xm∂xs. We first prove Theorem 1.9(i). Let f ∈

hΦω, z(Ω)∩L2(Ω) and M be as in (6.1) satisfying M > nqω

2p−Φ

. Then by Lemma 6.3, we

know that there exist a sequence {ak}k of (ρ, 2, M)ω-atoms and a sequence {λk}kof numbers such that

(6.5) f =∑k

λkak



and

(6.6) ‖f‖hΦω, z(Ω) ∼ Λ ({λkak}k) ,

where Λ ({λkak}k) is as in Definition 6.2.To finish the proof of Theorem 1.9(i), we need to show that for any (ρ, 2, M)ω-

atom a supported in the ball B(x0, r0) ∩ Ω and any λ ∈ C,

(6.7)

∫Ω

Φ(T (λa)(x))ω(x) dx � ω(BΩ

0

)Φ

(|λ|

ω(BΩ0 )ρ(ω(B

Ω0 ))

),

where and in what follows, BΩ0 := B(x0, r0) ∩ Ω.

Indeed, if (6.7) holds, then by (6.5) and the assumption that Φ is subadditive,we know that for all λ ∈ (0,∞),∫

Ω

Φ

(T

(f

λ

)(x)

)ω(x) dx ≤

∑k

∫Ω

Φ

(T

(λkakλ

)(x)

)ω(x) dx

�∑k

ω(BΩ

k

)Φ

(|λk|

λω(BΩk )ρ(ω(B

Ωk ))

),

where for each k , BΩk := B(xk, rk)∩Ω and supp ak ⊂ B(xk, rk)∩Ω, which, together

with (6.6), implies that‖T (f)‖LΦ

ω(Ω) � ‖f‖hΦω, z(Ω).

From this, the fact that hΦω, z(Ω)∩L2(Ω) is dense in hΦ

ω, z(Ω) and a density argument,we deduce that Theorem 1.9(i) holds.

Now we prove (6.7) by considering the following two cases for r0.

Case i) r0 ≥ 1. In this case, the proof of (6.7) is similar to the proof of (5.4). Weomit the details.

Case ii) r0 < 1. In this case, we have∫Ω

Φ(T (λa)(x))ω(x) dx =∞∑j=0

∫UΩ

j (B0)

Φ(T (λa)(x))ω(x) dx =:∞∑j=0

Ij .

For j ∈ {0, 1, 2}, similar to the estimate of (5.6), we know that

(6.8) Ij � ω(BΩ

0

)Φ

(|λ|

ω(BΩ0 )ρ(ω(B

Ω0 ))

).

Now we deal with Ij for j ∈ N with j ≥ 3. Take q ∈ (1, 2) such that ω ∈ RHq′(Rn),

where 1q + 1

q′ = 1. Then from Jensen’s inequality, Holder’s inequality, and Lemma

2.2(v), we deduce that

Ij ≤ ω(UΩj (B0)

)Φ

⎛⎝ 1

ω(UΩj (B0))

{∫UΩ

j (B0)

|T (λa)(x)|q dx} 1

q

(6.9)

×{∫

UΩj (B0)

[ω(x)]q′dx

} 1q′⎞⎠

� ω(UΩj (B0)

)Φ

⎛⎝ 1

|UΩj (B0)|

1q

{∫UΩ

j (B0)

|T (λa)(x)|q dx} 1

q

⎞⎠ .



Moreover, since

T (λa) = 2

∫ ∞

0

∂2e−2tL(λa)

∂xm∂xsdt,

we conclude that for each j ∈ N with j ≥ 3,

‖T (λa)‖Lq(UΩj (B0))(6.10)

≤ 2

∥∥∥∥∥∫ r20

0

∂2e−2tL(λa)

∂xm∂xsdt

∥∥∥∥∥Lq(UΩ

j (B0))

+ 2

∥∥∥∥∥∫ [ diam (Ω)]2

r20

· · · dt∥∥∥∥∥Lq(UΩ

j (B0))

+ 2

∥∥∥∥∥∫ ∞

[ diam (Ω)]2· · · dt

∥∥∥∥∥Lq(UΩ

j (B0))

=: I + II + III.

We first estimate I. By j ≥ 3, we know that dist (UΩj (B0), B

Ω0 ) ≥ 2j−2r0. From

this and Minkowski’s inequality, Lemma 3.9 and (6.2), we infer that∥∥∥∥∂2e−2tL(λa)

∂xm∂xs

∥∥∥∥Lq(UΩ

j (B0))

≤∫BΩ

0

[∫UΩ

j (B0)

|∇2xKt(x, y)|q dx

] 1q

|λa(y)| dy

� |λ|‖a‖L1(Ω)tn2 ( 1

q−1)−1e−γ[2jr0]2

t ,

which, together with Minkowski’s inequality, implies that, for any M0 ∈ (n2 (1 −1q ),∞),

I � |λ|‖a‖L1(Ω)

∫ r20

0

e−γ[2jr0]2

t tn2 ( 1

q−1)−1 dt(6.11)

� |λ|‖a‖L2(Ω)|BΩ0 |

12

∫ r20

0

(t

[2jr0]2

)M0

tn2 ( 1

q−1)−1 dt

� 2−2jM0|λ||B0|

1q

ω(BΩ0 )ρ(ω(B

Ω0 ))

.

Now we deal with II. Pick M1 ∈ (nqω2p−

Φ

,M). By a = LMb, the fact that for each

k ∈ N,

(−1)kLke−tL =dk

dtke−tL,

Lemma 3.9 and (6.3), we conclude that

II � |λ|∫ [ diam (Ω)]2

r20

∥∥∥∥ ∂2

∂xm∂xs

(dM

dtMe−2tLb

)∥∥∥∥Lq(UΩ

j (B0))

dt(6.12)

� |λ|∫ [ diam (Ω)]2

r20

∫BΩ

0

[∫UΩ

j (B0)

∣∣∣∣∇2x

(dM

dtMKt

)(x, y)

∣∣∣∣q dx

] 1q

|b(y)| dy dt

� |λ|‖b‖L1(Ω)

∫ [ diam (Ω)]2

r20

e−γ[2jr0]2

t tn2 ( 1

q−1)−(M+1) dt

� |λ|‖b‖L2(Ω)|BΩ0 |

12

∫ [ diam (Ω)]2

r20

(t

[2jr0]2

)M1

tn2 ( 1

q−1)−(M+1) dt



� 2−2jM1|λ||B0|

1q

ω(BΩ0 )ρ(ω(B

Ω0 ))

.

For III, similar to (6.12), we have

III � |λ|∫ ∞

[ diam (Ω)]2

∥∥∥∥ ∂2

∂xm∂xs

(dM

dtMe−2tLb

)∥∥∥∥Lq(UΩ

j (B0))

dt(6.13)

� |λ|∫ ∞

[ diam (Ω)]2

∫BΩ

0

[∫UΩ

j (B0)

∣∣∣∣∇2x

(dM

dtMKt

)(x, y)

∣∣∣∣q dx

] 1q

|b(y)| dy dt

� |λ|‖b‖L1(Ω)

∫ ∞

[ diam (Ω)]2e−γ

[2jr0]2

t t−(M+1) dt

� |λ|‖b‖L2(Ω)|BΩ0 |

12

∫ ∞

[ diam (Ω)]2

(t

[2jr0]2

)M1

t−(M+1) dt

� 2−2jM1|λ||B0|r2(M−M1)

0

ω(BΩ0 )ρ(ω(B

Ω0 ))

� 2−2jM1|λ||B0|

1q

ω(BΩ0 )ρ(ω(B

Ω0 ))

,

where we used the assumption that Ω is bounded and B(y,√t) ∩ Ω = Ω for all

t ∈ ([ diam (Ω)]2,∞) in the third inequality, and the assumption that rB < 1 in thelast inequality.

By M0 > nqω2pΦ

, we know that there exist p0 ∈ (0, p−Φ) and q0 ∈ (qω,∞) such that

Φ is of lower type p0 and M0 > nq02p0

. Thus, from (6.9), (6.10), (6.11), (6.12), (6.13),

Lemma 2.2(iii), the lower type p0 property of Φ and M0 > nq02p0

, we deduce that

∞∑j=3

Ij �∞∑j=3

ω(UΩj (B0)

)Φ

(2−2jM1 |λ||UΩ

j (B0)|1q

|B0|1q

ω(BΩ0 )ρ(ω(B

Ω0 ))

)

�∞∑j=3

2jnq02−(2M1+n/q)jp0ω(BΩ

0

)Φ

(|λ|

ω(BΩ0 )ρ(ω(B

Ω0 ))

)

� ω(BΩ

0

)Φ

(|λ|

ω(BΩ0 )ρ(ω(B

Ω0 ))

),

which, together with (6.8), implies that (6.7) holds in this case. This finishes theproof of Theorem 1.9(i).

Now we prove Theorem 1.9(ii). For each y ∈ Ω, let V (·, y) be the solution of theNeumann problem

(6.14)

{ΔV (·, y) = |Ω|−1, in Ω,

∂ν(x)[V (x, y)] = ∂ν(x)[Γ(x− y)], for x ∈ ∂Ω,

where ν(x) denotes the outward unit normal to ∂Ω at x ∈ ∂Ω. Then a convenientway of expressing the Green function GN for L = −Δ with the Neumann boundarycondition on Ω (namely, the integral kernel of the Neumann Green operator GN ) is

GN (x, y) = Γ(x− y)− V (x, y), x, y ∈ Ω, x �= y.



The Neumann problem (1.3) has a unique solution, up to an additive constant,given by

GN (f)(x) =

∫Ω

GN (x, y)f(y) dy(6.15)

=

∫Ω

Γ(x− y)f(y) dy −∫Ω

V (x, y)f(y) dy

=: E(f)(x)− V (f)(x),

where f ∈ C∞(Ω) satisfies that ∫Ω

f(y) dy = 0.

Let f ∈ hΦω, z(Ω)∩L2(Ω). Then by Lemma 6.3, we know that there exist a sequence

{ak}k of (ρ, 2, M)ω-atoms and a sequence {λk}k of numbers such that (6.5) and(6.6) hold.

To finish the proof of Theorem 1.9(ii), we only need to show that for any(ρ, 2, M)ω-atom a supported in the ball B(x0, r0) ∩ Ω and any λ ∈ C,

(6.16)

∫Ω

Φ([T (λa)]+Ω, ϕ(x)

)ω(x) dx � ω(BΩ

0 )Φ

(|λ|

ω(BΩ0 )ρ(ω(B

Ω0 ))

),

where and in what follows, BΩ0 := B(x0, r0) ∩ Ω, and [T (λa)]+Ω, ϕ is as in (1.7).

Indeed, if (6.16) holds, then by (6.5) and the assumption that Φ is subadditive,we conclude that for all λ ∈ (0,∞),

∫Ω

Φ

([T

(f

λ

)]+Ω, ϕ

(x)

)ω(x) dx ≤

∑k

∫Ω

Φ

([T

(λkakλ

)]+Ω, ϕ

(x)

)ω(x) dx

�∑k

ω(BΩk )Φ

(|λk|

λω(BΩk )ρ(ω(B

Ωk ))

),

where for each k, BΩk := B(xk, rk)∩Ω and supp ak ⊂ B(xk, rk)∩Ω, which, together

with Theorem 1.7 and (6.6), implies that

‖T (f)‖hΦω, r(Ω) � ‖f‖hΦ

ω, z(Ω).

From this, the fact that hΦω, z(Ω)∩L2(Ω) is dense in hΦ

ω, z(Ω) and a density argument,we deduce that Theorem 1.9(ii) holds.

Now we prove (6.16). By (6.15), it suffices to show

∫Ω

Φ

([∂2E(λa)

∂xm∂xs

]+Ω, ϕ

(x)

)ω(x) dx+

∫Ω

Φ

([∂2V (λa)

∂xm∂xs

]+Ω, ϕ

(x)

)ω(x) dx

� ω(BΩ

0

)Φ

(|λ|

ω(BΩ0 )ρ(ω(B

Ω0 ))

).



By the fact that L conserves probability, namely, e−tL1 = 1 for all t ∈ (0,∞), weknow that ∫

Ω

a(x) dx = 0

(see also [26, (5.4)]). Then, similar to the proof of [86, Theorem 8.2], we have∫Ω

Φ

([∂2E(λa)

∂xm∂xs

]+Ω, ϕ

(x)

)ω(x) dx ∼

∥∥∥∥∂2E(λa)

∂xm∂xs

∥∥∥∥hΦω, r(Ω)

� ‖λa‖hΦω, r(Ω).(6.17)

Moreover, by (6.14) and (6.15), we see that ∂2V (λa)∂xm∂xs

is harmonic in Ω. Hence, anapplication of Theorem 1.7, in which we take the function ϕ to be radial, yields, onaccount of the mean value property for harmonic functions, that for all x ∈ Ω,(

∂2V (λa)

∂xm∂xs

)+

Ω, ϕ

(x) =

∣∣∣∣∂2V (λa)

∂xm∂xs(x)

∣∣∣∣ ,which, together with Theorems 1.7 and 1.9(i), and Lemma 6.3, implies that∥∥∥∥∂2V (λa)

∂xm∂xs

∥∥∥∥hΦω, r(Ω)

∼∥∥∥∥∥(∂2V (λa)

∂xm∂xs

)+

Ω, ϕ

∥∥∥∥∥LΦ

ω(Ω)

∼∥∥∥∥∂2V (λa)

∂xm∂xs

∥∥∥∥LΦ

ω(Ω)

�∥∥∥∥∂2E(λa)

∂xm∂xs

∥∥∥∥LΦ

ω(Ω)

+

∥∥∥∥∂2GN (λa)

∂xm∂xs

∥∥∥∥LΦ

ω(Ω)

� ‖λa‖hΦω, r(Ω).

This, combined with (6.17) and Lemma 6.3, implies that (6.16) holds, which com-pletes the proof of Theorem 1.9. �

Acknowledgements

The authors would like to thank Professor Lixin Yan for some suggestive discus-sions on the topic of this paper.

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School of Mathematical Sciences, Beijing Normal University, Laboratory of Math-

ematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic

of China

E-mail address: [email protected]

Department of Mathematics and Department of Computer Science, Georgetown Uni-

versity, Washington, DC 20057




of China




of China





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