TRACE IDEAL PROPERTIES OF A CLASS OF INTEGRAL OPERATORS · type operators in Theorem 3.4. Finally,...
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TRACE IDEAL PROPERTIES OF A CLASS
OF INTEGRAL OPERATORS
FRITZ GESZTESY AND ROGER NICHOLS
Dedicated with admiration to Emma Previato – Geometer Extraordinaire
Abstract. We consider a particular class of integral operators Tγ,δ in L2(Rn),
n ∈ N, n > 2, with integral kernels Tγ,δ( · , · ) bounded (Lebesgue) a.e. by
Tγ,δ(x, y) 6 C〈x〉−δx− y2γ−n〈y〉−δ, x, y ∈ Rn, x 6= y,for fixed C ∈ (0,∞), 0 < 2γ < n, δ > γ, and prove that
Tγ,δ ∈ Bp(L2(Rn)
)for p > n/(2γ), p > 2.
(Here 〈x〉 := (1 + x2)1/2, x ∈ Rn, and Bp abbreviates the `pbased traceideal.) These integral operators (and their matrixvalued analogs) naturallyarise in the study of multidimensional Schrödinger and Diractype operators
and we describe an application to the case of massless Diractype operators.
1. Introduction
The principal aim of this paper is to derive trace ideal properties of a classof (matrixvalued) integral operators that naturally arise in the context of multidimensional Schrödinger and Diractype operators. More precisely, we will focuson integral operators Tγ,δ in L
2(Rn), n ∈ N, n > 2, which in the scalar context areassociated with integral kernels Tγ,δ( · , · ) that are bounded (Lebesgue) a.e. by
Tγ,δ(x, y) 6 C〈x〉−δx− y2γ−n〈y〉−δ, x, y ∈ Rn, x 6= y, (1.1)
for fixed C ∈ (0,∞), 0 < 2γ < n, δ > γ. We then prove in Theorem 3.1 that
Tγ,δ ∈ Bp(L2(Rn)
)for p > n/(2γ), p > 2. (1.2)
This result is then applied to prove Theorem 3.4 which derives a uniform traceideal norm bound with respect to the spectral parameter on the resolvent of massless Diractype operators if the Diractype resolvent is viewed as a map betweensuitable weighted L2(Rn) spaces. Such estimates are wellknown to imply limitingabsorption principles for the Diractype operator in question which, in turn, havestrong spectral implications (such as the absence of any singular continuous spectrum, etc.). Moreover, the fact that trace ideal bounds are involved can now beused to infer continuity properties of underlying spectral shift functions which yields
Date: July 20, 2020.2010 Mathematics Subject Classification. Primary: 46B70, 47B10, 47G10, 47L20; Secondary:
35Q40, 81Q10.Key words and phrases. Trace ideals, interpolation theory, massless Diractype operators.Published in Integrable Systems and Algebraic Geometry. Volume 1, R. Donagi and T. Shaska
(eds.), London Math. Soc. Lecture Note Ser. 458, Cambridge University Press, Cambridge, UK,
2020, pp. 13–37.
1

2 F. GESZTESY AND R. NICHOLS
further applications to the Witten index for a particular class of nonFredholm operators as discussed, for instance, in [3]–[9], [12], [24] (see also Remark 3.5). We notethat twodimensional massless Diractype operators are also known to be relevantin the context of graphene, one more reason to study the massless case.
In Section 2 we collect a fair amount of background material, some of which iscrucial for our main Section 3. In particular, we focus on integral operators withintegral kernels closely related to the righthand side of (1.1) and survey some of thepertinent literature in this context, including a fundamental criterion by Nirenbergand Walker [22] (we include a detailed proof of the latter), a result on absolutekernels, some wellknown Schur tests, and a trace norm estimate due to Demuth,Stollmann, Stolz, and van Casteren [10] (again we supply the proof of the latter).Section 2 also recalls a version of the Sobolev inequality and a fundamental traceideal interpolation result. Our principal Section 3 then proves the inclusion (1.2)in Theorem 3.1 and demonstrates its applicability to the case of massless Diractype operators in Theorem 3.4. Finally, Appendix A collects some useful resultson pointwise domination of linear operators and its consequences in connectionwith boundedness, compactness, and Hilbert–Schmidt properties. We include adiscussion of block matrix operator situations necessitated by the study of Diractype operators.
We conclude this introduction with some comments on the notation employed inthis paper: Let H be a separable complex Hilbert space, (·, ·)H the scalar productin H (linear in the second argument), and IH the identity operator in H.
Next, if T is a linear operator mapping (a subspace of) a Hilbert space intoanother, then dom(T ) and ker(T ) denote the domain and kernel (i.e., null space) ofT . The spectrum, point spectrum (the set of eigenvalues), the essential spectrum ofa closed linear operator in H will be denoted by σ( · ), σp( · ), σess( · ), respectively.Similarly, the absolutely continuous and singularly continuous spectrum of a selfadjoint operator in H are denoted by σac( · ) and σsc( · ).
The Banach spaces of bounded and compact linear operators on a separablecomplex Hilbert space H are denoted by B(H) and B∞(H), respectively; the corresponding `pbased trace ideals will be denoted by Bp(H), their norms are abbreviated by ‖ · ‖Bp(H), p ∈ [1,∞). Moreover, trH(A) denotes the corresponding traceof a trace class operator A ∈ B1(H).
If p ∈ [1,∞) ∪ {∞}, then p′ ∈ [1,∞) ∪ {∞} denotes its conjugate index, thatis, p′ := (1− 1/p)−1. If Lebesgue measure is understood, we simply write Lp(M),M ⊆ Rn measurable, n ∈ N, instead of the more elaborate notation Lp(M ; dnx).For x = (x1, . . . , xn) ∈ Rn, n ∈ N, we abbreviate 〈x〉 := (1 + x2)1/2.
Finally, b · c denotes the floor function on R, that is, bxc characterizes the largestinteger less than or equal to x ∈ R.
2. Some Background Material
This preparatory section is primarily devoted to various results of integral operators, but we also briefly recall Sobolev’s inequality and some interpolation resultsfor trace ideal operators.
We start by recalling the following version of Sobolev’s inequality (see, e.g., [26,Corollary I.14]), to be employed in the proof of Theorem 3.1.

ON A CLASS OF INTEGRAL OPERATORS 3
Theorem 2.1. Let n ∈ N, r, s ∈ (1,∞), 0 < λ < n, r−1 + s−1 + λn−1 = 2,f ∈ Lr(Rn), h ∈ Ls(Rn). Then, there exists Cr,s,λ,n ∈ (0,∞) such thatˆ
Rn×Rndnx dny
f(x)h(y)x− yλ
6 Cr,s,λ,n‖f‖Lr(Rn)‖h‖Ls(Rn). (2.1)
We continue this section with a special case of a very interesting result of Nirenberg and Walker [22] also to be employed in the proof of Theorem 3.1. For convenience of the reader we offer a detailed proof.
Theorem 2.2. Let n ∈ N, c, d ∈ R, c+ d > 0, and consider
Kc,d(x, y) = x−cx− y(c+d)−ny−d, x, y ∈ Rn, x 6= x′. (2.2)
Then the integral operator Kc,d in L2(Rn) with integral kernel Kc,d( · , · ) in (2.2)
is bounded if and only if
c < n/2 and d < n/2. (2.3)
Proof. To prove the necessity of the conditions (2.3), assume K ∈ B(L2(Rn)). ThenˆRndny Kc,d( · , y)f(y) ∈ L2(Rn), f ∈ L2(Rn). (2.4)
In particular, choosing f = χBn(0;1)
, the characteristic function of the closed unit
ball in Rn,Bn(0; 1) = {x ∈ Rn  x 6 1}, (2.5)
which has finite Lebesgue measure,∣∣Bn(0; 1)∣∣ = πn/2/Γ((n/2) + 1), one obtainsˆ
Bn(0;1)
dny Kc,d( · , y) ∈ L2(Rn). (2.6)
Since Kc,d( · , · ) is symmetric in x and y, one also infersˆBn(0;1)
dnxKc,d(x, · ) ∈ L2(Rn). (2.7)
In summary, if Kc,d ∈ B(L2(Rn)), thenˆBn(0;1)
dny Kc,d( · , y) ∈ L2(Rn) andˆBn(0;1)
dnxKc,d(x, · ) ∈ L2(Rn). (2.8)
To investigate the behavior ofˆBn(0;1)
dny
xcx− yn−c−dyd, x ∈ Rn\{0}, (2.9)
as x → ∞, one writesˆBn(0;1)
dny
xcx− yn−c−dyd=
1
xn−d
ˆBn(0;1)
dny∣∣∣∣ xx − yx∣∣∣∣n−c−dyd
,
x ∈ Rn\{0}. (2.10)
If x, y ∈ Rn with x > 2 and y 6 1, then the elementary estimates1
26 1− 1
x6
∣∣∣∣ xx − yx∣∣∣∣ 6 1 + 1x 6 32 (2.11)

4 F. GESZTESY AND R. NICHOLS
imply
C1 6
∣∣∣∣ xx − yx∣∣∣∣c+d−n 6 C2, x, y ∈ Rn, x > 2, y 6 1, (2.12)
for some constants C1, C2 ∈ (0,∞). In particular, the finiteness of the integral in(2.9) implies the finiteness of the integralˆ
Bn(0;1)
dny
yd. (2.13)
By Lebesgue’s dominated convergence theorem,
limx→∞
ˆBn(0;1)
dny∣∣∣∣ xx − yx∣∣∣∣n−c−dyd
=
ˆBn(0;1)
dny limx→∞
1∣∣∣∣ xx − yx∣∣∣∣n−c−dyd
=
ˆBn(0;1)
dny
yd=: I1,n,d, (2.14)
since (2.11) implies
limx→∞
∣∣∣∣ xx − yx∣∣∣∣ = 1, y ∈ Rn. (2.15)
Therefore, by (2.10) and (2.14),ˆBn(0;1)
dny
xcx− yn−c−dyd∼ I1,n,d ·
1
xn−das x → ∞, (2.16)
and, similarly,ˆBn(0;1)
dnx
xcx− yn−c−dyd∼ I1,n,c ·
1
yn−cas y → ∞. (2.17)
In light of (2.16) and (2.17), the containments in (2.8) hold only if (n − d)2 > nand (n− c)2 > n, that is, only if c < n/2 and d < n/2.
To prove sufficiency of the conditions in (2.3), assume c < n/2 and d < n/2. Itsuffices to prove the claim Kc,d ∈ B(L2(Rn)) in the special case where c, d ∈ [0,∞).The claim for general c and d then follows from this special case. Indeed, if c werenegative, for example, then d would be positive, and the elementary inequality
xx− y
6 1 +yx− y
, x, y ∈ Rn, x 6= y, (2.18)
implies
Kc,d(x, y) =
(xx− y
)−c1
x− yn−dyd6M−c
(1 +
y−c
x− y−c
)1
x− yn−dyd
=M−c
x− yn−dyd+
M−cx− yn−(c+d)yc+d
, x, y ∈ Rn\{0}, x 6= y, (2.19)
where for each α ∈ [0,∞), Mα ∈ (0,∞) is a constant such that
(1 + t)α 6Mα(1 + tα), t ∈ [0,∞). (2.20)
Note that the existence of Mα is guaranteed by the fact that, for each fixed α ∈[0,∞), the function
φα(t) =(1 + t)α
1 + tα, t ∈ [0,∞), (2.21)

ON A CLASS OF INTEGRAL OPERATORS 5
is continuous on [0,∞) and has a finite limit as t → ∞. The special case underconsideration (viz., c, d ∈ [0,∞)) then implies that the righthand side of (2.19) isthe sum of the kernels of two integral operators in B(L2(Rn)), so that Kc,d( · , · )generates a bounded operator on L2(Rn). Therefore, for the remainder of thisproof, we assume 0 6 c < n/2 and 0 6 d < n/2.
By the arithmeticgeometric mean inequality,
x >n∏j=1
xj 1/n, x = (x1, . . . , xn) ∈ Rn, (2.22)
implying
Kc,d(x, y) 6n∏j=1
1
xj c/nxj − yj 1−(c+d)/nyj d/n, (2.23)
x = (xj)nj=1, y = (yj)
nj=1 ∈ Rn, xj 6= yj , xj 6= 0, yj 6= 0, 1 6 j 6 n.
Therefore, by Lemma A.4, it suffices to show that the integral operator Jc,d withintegral kernel
Jc,d(s, t) =1
sc/ns− t1−(c+d)/ntd/n, s, t ∈ R\{0}, s 6= t, (2.24)
belongs to B(L2(R)).The function Jc,d( · , · ) defined by (2.24) is homogeneous of degree (−1). In
addition,ˆ ∞0
ds J(s, 1) s−1/2 =
ˆ ∞0
ds
s(1/2)+(c/n)s− 11−(c+d)/n

6 F. GESZTESY AND R. NICHOLS
Theorem 2.3. Let n ∈ N and β ∈ (0, n). Suppose p0, q0, p, q ∈ (1,∞) with1
p′+
1
p0< 1,
1
q+
1
q0< 1,
1
p=
1
q+
1
p0+
1
q0− βn. (2.28)
If
a ∈ Lp0(Rn) and b ∈ Lq0(Rn), (2.29)then the kernel
k(x, y) = a(x)x− yβ−nb(y) for a.e. x, y ∈ Rn (2.30)
generates a bounded integral operator K ∈ B(Lq(Rn), Lp(Rn)).
While Theorem 2.3 permits a variety of functions a and b, it does not apply tothe kernel Kc,d in (2.2) due to the integrability requirements in (2.29).
Theorem 2.2 gives necessary and sufficient conditions for boundedness of theintegral operator Kc,d. In general, there are no known practical necessary andsufficient conditions for the boundedness of an integral operator. However, thereare various sufficient conditions which allow one to infer boundedness of an integraloperator from appropriate bounds on the integral kernel itself. Illustrative examplesare the wellknown Schur criteria or Schur tests to which we briefly turn next forthe sake of completeness.
The following wellknown version of the Schur test (cf., e.g., [16, Theorem 5.2])provides a sufficient condition for boundedness between L2spaces in terms of pointwise bounds on the integral kernel when integrated against a pair of measurabletrial functions.
Theorem 2.4 (Schur test–first version). Let (X,M, dµ) and (Y,N , dν) be σfinitemeasure spaces and k : X × Y → [0,∞) a measurable function. If φ : X → (0,∞)and ψ : Y → (0,∞) are measurable and if α, β ∈ (0,∞) are such thatˆ
Y
dν(y) k(x, y)ψ(y) 6 αφ(x) for a.e. x ∈ X (2.31)
and ˆX
dµ(x) k(x, y)φ(x) 6 αψ(y) for a.e. y ∈ Y , (2.32)
then k is the integral kernel of a bounded integral operator
K ∈ B(L2(Y ; dν), L2(X; dµ)
)(2.33)
and
‖K‖B(L2(Y ;dν),L2(X;dµ)) 6 (αβ)1/2. (2.34)
Proof. If f ∈ L2(Y ; dν), then using the Cauchy–Schwarz inequality one obtainsˆX
dµ(x)
(ˆY
dν(y) k(x, y)f(y))
=
ˆX
dµ(x)
(ˆY
dν(y) k(x, y)1/2ψ(y)1/2[k(x, y)
ψ(y)
]1/2f(y)
)26ˆX
dµ(x)
(ˆY
dν(y k(x, y)ψ(y))
)( ˆY
dν(y′)k(x, y′)
ψ(y′)f(y′)2
)6ˆX
dµ(x)αφ(x)
(ˆY
dν(y)k(x, y)
ψ(y)f(y)2
)

ON A CLASS OF INTEGRAL OPERATORS 7
= α
ˆY
dν(y)f(y)2
ψ(y)
(ˆX
dµ(x) k(x, y)φ(x)
)6 αˆY
dν(y)f(y)2
ψ(y)βψ(y)
= αβ
ˆY
dν(y) f(y)2. (2.35)
�
Example 2.5 (Abel kernel). The integral operator K in L2((0, 1)) generated bythe kernel
k(x, y) =
{0, x 6 y,
(x− y)−1/2, y < x,(2.36)
belongs to B(L2((0, 1))). In fact, the Schur test applies with ψ(x) = φ(x) = 1 fora.e. x ∈ (0, 1) and α = β = 2.
The proof of the following Lpbased version of the Schur test relies on Hölder’sinequality (cf., e.g., [29, Satz 6.9]).
Theorem 2.6 (Schur test–second version). Let p, p′ ∈ (1,∞) with p−1+(p′)−1 = 1and let (X,M, dµ) and (Y,N , dν) be σfinite measure spaces. Suppose k : X×Y →C is a measurable function and that there exist measurable functions k1, k2 : X ×Y → [0,∞) such that
k(x, y) 6 k1(x, y)k2(x, y) for a.e. (x, y) ∈ X × Y , (2.37)
and
‖k1(x, · )‖Lp′ (Y ;dν) 6 C1, ‖k2( · , y)‖Lp(X;dµ) 6 C2, (2.38)
for µa.e. x ∈ X and νa.e. y ∈ Y for some constants C1, C2 ∈ (0,∞). Then k isthe integral kernel of a bounded integral operator
K ∈ B(Lp(Y ; dν), Lp(X; dµ)
)(2.39)
and
‖K‖B(Lp(Y ;dν),Lp(X;dµ)) 6 C1C2. (2.40)
While Theorems 2.4 and 2.6 provide useful sufficient conditions for an integraloperator to be bounded over an Lpspace, they do not yield information aboutpossible compactness or trace ideal properties of the integral operator. (For compactness properties, see, e.g., [16, § 13, 14], [18, § 11], [20, Ch. 2], [28, Sect. 6.3],[32, Ch. V].) In particular, neither Theorem 2.4 nor Theorem 2.6 implies the traceideal property 1.2. As an example of a result which provides sufficient conditionsfor an integral operator to belong to the trace class, we mention the following resulton general integral operators due to [10] and provide its short proof.
Theorem 2.7. Let (X,A, µ) be a σfinite measure space and suppose that A( · , · ),B( · , · ) : X ×X → C are measurable such that
A( · , x), B(x, · ) ∈ L2(X; dµ) for a.e. x ∈ X,ˆX
dµ(y) ‖A( · , y)‖L2(X;dµ)‖B(y, · )‖L2(X;dµ)

8 F. GESZTESY AND R. NICHOLS
Then there exists a trace class operator AB : L2(X; dµ)→ L2(X; dµ) with integralkernel
AB(x, y) =
ˆX
dµ(t)A(x, t)B(t, y), (2.42)
such that
‖AB‖B1(L2(X;dµ)) 6 ‖A( · , y)‖L2(X;dµ)‖B(y, · )‖L2(X;dµ). (2.43)
Proof. Introducing g(y) := ‖B(y, · )‖L2(X;dµ), h(y) := ‖A( · , y)‖L2(X;dµ), and employing the (unusual) convention g(x)−1 = 0 if g(x) = 0, we denote by Mf themaximally defined operator of multiplication by f in the space L2(X; dµ). Then
AB = AMh−1M(hg)1/2M(hg)1/2Mg−1B, (2.44)
and AMh−1M(hg)1/2 and M(hg)1/2Mg−1B are seen to be Hilbert–Schmidt operators.For instance,
‖AMh−1M(hg)1/2‖2B2(L2(X;dµ)) =ˆX
dµ(x)
ˆX
dµ(y)∣∣A(x, y)h(y)−1(hg)(x)1/2∣∣2
=
ˆX
dµ(x)
ˆX
dµ(y)∣∣A(x, y)h(y)−1/2(g)(x)1/2∣∣2
=
ˆX
dµ(y) g(y)h(y) n/(2γ) in (1.2). Tocircumvent these difficulties, we shall employ interpolation methods in Section 3 toprove (1.2). In particular, we will make use of the following trace ideal interpolationresult, see, for instance, [14, Theorem III.13.1], [31, Theorem 0.2.6] (see also [13],[15, Theorem III.5.1]) in the proof of (1.2) (cf. Theorem 3.1).
Theorem 2.8. Let pj ∈ [1,∞) ∪ {∞}, Σ = {ζ ∈ C Re(ζ) ∈ (ξ1, ξ2)}, ξj ∈ R,ξ1 < ξ2, j = 1, 2. Suppose that A(ζ) ∈ B(H), ζ ∈ Σ and that A( · ) is analytic onΣ, continuous up to ∂Σ, and that ‖A( · )‖B(H) is bounded on Σ. Assume that forsome Cj ∈ (0,∞),
supη∈R‖A(ξj + iη)‖Bpj (H) 6 Cj , j = 1, 2. (2.46)
Then
A(ζ) ∈ Bp(Re(ζ))(H),1
p(Re(ζ))=
1
p1+
Re(ζ)− ξ1ξ2 − ξ1
[1
p2− 1p1
], ζ ∈ Σ, (2.47)
and
‖A(ζ)‖Bp(Re(ζ))(H) 6 C(ξ2−Re(ζ))/(ξ2−ξ1)1 C
(Re(ζ)−ξ1)/(ξ2−ξ1)2 , ζ ∈ Σ. (2.48)
In case pj =∞, B∞(H) can be replaced by B(H).

ON A CLASS OF INTEGRAL OPERATORS 9
In the next section, we shall employ Theorem 2.8 to interpolate between theB(L2(Rn)) and Bp(L2(Rn)) properties for a family of integral operators Tγ,δ inL2(Rn), n > 2, with kernels bounded in absolute value according to (1.1), forappropriate values of the parameters γ, δ.
3. Interpolation and Trace Ideal Propertiesof a Class of Integral Operators
In this section we combine Theorems 2.1, 2.2, 2.8, and an interpolation procedureto prove Theorem 3.1 below. The latter asserts a trace ideal containment for integraloperators in L2(Rn), n > 2, with kernels bounded in absolute value by a constanttimes a function of the form 〈x〉−δx−y2γ−n〈y〉−δ, x, y ∈ Rn, x 6= y, for appropriatevalues of the parameters γ, δ. Theorem 3.4 then provides an application of Theorem3.1 to the case of ndimensional massless Diractype operators.
A combination of Theorems 2.1, 2.2, and 2.8 yields the following general result.
Theorem 3.1. Let n ∈ N, n > 2, 0 < 2γ < n, δ > γ, and suppose that Tγ,δ is anintegral operator in L2(Rn) whose integral kernel Tγ,δ( · , · ) satisfies the estimate
Tγ,δ(x, y) 6 C〈x〉−δx− y2γ−n〈y〉−δ, x, y ∈ Rn, x 6= y (3.1)
for some C ∈ (0,∞). Then,
Tγ,δ ∈ Bp(L2(Rn)
), p > n/(2γ), p > 2, (3.2)
and
‖Tγ,δ‖Bn/(2γ−ε)(L2(Rn)) 6 supη∈R
[‖Tγ,δ(−2γ + ε+ iη)‖B(L2(Rn))
]2[−2γ+(n/2)+ε]/n× supη∈R
[‖Tγ,δ(−2γ + (n/2) + ε+ iη)‖B2(L2(Rn))
]2(2γ−ε)/n(3.3)
for 0 < ε sufficiently small.
Proof. Following the idea behind Yafaev’s proof of [31, Lemma 0.13.4], we introducethe analytic family of integral operators Tγ,δ( · ) in L2(Rn) generated by the integralkernel
Tγ,δ(ζ;x, y) = Tγ,δ(x, y) 〈x〉−(ζ/2)x− yζ〈y〉−(ζ/2), x, y ∈ Rn, x 6= y, (3.4)
noting Tγ,δ(0) = Tγ,δ. By Theorems 2.2 and A.2 (i) (for N = 1),
Tγ,δ(ζ) ∈ B(L2(Rn)
), 0 < Re(ζ) + 2γ < n, δ > γ. (3.5)
To check the Hilbert–Schmidt property of Tγ,δ( · ) one estimates for the square ofTγ,δ( · ; · , · ),
Tγ,δ(ζ;x, y)2 6 〈x〉−2δ−Re(ζ)x− y2Re(ζ)+4γ−2n〈x〉−2δ−Re(ζ),x, y ∈ Rn, x 6= y,
(3.6)
and hence one can apply Theorem 2.1 upon identifying λ = 2n − 4γ − 2Re(z),r = s = n/[Re(ζ) + 2γ], and f = h = 〈 · 〉−[2δ+Re(ζ)], to verify that 0 < λ < ntranslates into n/2 < Re(ζ) + 2γ < n, and f ∈ Lr(Rn) holds with r ∈ (1, 2) ifδ > γ. Hence,
Tγ,δ(ζ) ∈ B2(L2(Rn)
), n/2 < Re(ζ) + 2γ < n, δ > γ. (3.7)

10 F. GESZTESY AND R. NICHOLS
It remains to interpolate between the B(L2(Rn)
)and B2
(L2(Rn)
)properties, em
ploying Theorem 2.8 as follows. Choosing 0 < ε sufficiently small, one identifiesξ1 = −2γ + ε, ξ2 = −2γ + (n/2) + ε, p1 =∞, p2 = 2, and hence obtains
p(Re(ζ)) = n/[Re(ζ) + 2γ − ε], (3.8)
in particular, p(0) > n/(2γ) (and of course, p(0) > 2). Since ε may be takenarbitrarily small, (3.2) follows from (3.8) and (3.3) is a direct consequence of (2.48).
�
While subordination in general only applies to Bpideals with p even (see thediscussion in [27, p. 24 and Addendum E]), the use of complex interpolation inTheorem 3.1 (and the focus on bounded and Hilbert–Schmidt operators) permitsone to avoid this restriction.
Theorem 3.1 represents the principal result of this paper and to the best of ourknowledge it appears to be new.
The singularity structure on the diagonal of the integral kernels Kc,d introducedin (2.2) naturally matches the one of multidimensional Schrödinger and Diractypeoperators as we will indicate next.
As a brief preparation we first record the asymptotic behavior of Hankel func
tions of the first kind with index ν > 0 (cf. e.g., [1, Sect. 9.1]), H(1)ν ( · ), as thelatter are crucial in the context of constant coefficient (i.e., free, or noninteracting)Schrödinger and Diractype operators, a natural first step in studying Schrödingerand Diractype operators with nontrivial interaction terms (i.e., potentials). Employing, for instance, [1, p. 360, 364], one obtains
H(1)0 (ζ) =
ζ→0(2i/π)ln(ζ) +O
(ln(ζ)ζ2
), (3.9)
H(1)ν (ζ) =ζ→0−(i/π)2νΓ(ν)ζ−ν +
{O(ζmin(ν,−ν+2)
), ν /∈ N,
O(ln(ζ)ζν
)+O
(ζ−ν+2
), ν ∈ N,
(3.10)
Re(ν) > 0,
H(1)ν (ζ) =ζ→∞
(2/π)1/2ζ−1/2eiζ−(νπ/2)−(π/4), ν > 0, Im(ζ) > 0. (3.11)
Starting with the Laplacian in L2(Rn),
h0 = −∆, dom(h0) = H2(Rn), (3.12)
the Green’s function of h0, denoted by g0(z; · , · ), is then of the form,
g0(z;x, y) := (h0 − zI)−1(x, y)
=
(i/4)
(2πz−1/2x− y
)(2−n)/2H
(1)(n−2)/2
(z1/2x− y
), n > 2, z ∈ C\{0},
1
(n− 2)ωn−1x− y2−n, n > 3, z = 0,
z ∈ C\[0,∞), Im(z1/2
)> 0, x, y ∈ Rn, x 6= y, (3.13)
where ωn−1 = 2πn/2/Γ(n/2) (Γ( · ) the Gamma function, cf., e.g., [1, Sect. 6.1])
represents the area of the unit sphere Sn−1 in Rn.

ON A CLASS OF INTEGRAL OPERATORS 11
As z → 0, g0(z; · , · ) is continuous on the offdiagonal for n > 3,
limz→0
g0(z;x, y) = g0(0;x, y) =1
(n− 2)ωn−1x− y2−n,
x, y ∈ Rn, x 6= y, n ∈ N, n > 3,(3.14)
but blows up for n = 2 as
g0(z;x, y) =z→0− 1
2πln(z1/2x− y/2
)[1 +O
(zx− y2
)]+
1
2πψ(1)
+O(zx− y2
), x, y ∈ R2, x 6= y.
(3.15)
Here ψ(w) = Γ′(w)/Γ(w) denotes the digamma function (cf., e.g., [1, Sect. 6.3]).This briefly illustrates the relevance of the diagonal singularity structure x −y(c+d)−n in Kc,d in (2.2).
To describe an application to massless Dirac operators we need additional preparations. To rigorously define the free massless ndimensional Dirac operators to bestudied in the sequel, we now introduce the following set of basic hypotheses assumed for the remainder of this section.
Hypothesis 3.2. Let n ∈ N, n > 2.(i) Set N = 2b(n+1)/2c and let αj, 1 6 j 6 n, αn+1 := β, denote n + 1 anticommuting Hermitian N ×N matrices with squares equal to IN , that is,
α∗j = αj , αjαk + αkαj = 2δj,kIN , 1 6 j, k 6 n+ 1. (3.16)
Here IN denotes the N ×N identity matrix.(ii) Introduce in [L2(Rn)]N the free massless Dirac operator
H0 = α · (−i∇) =n∑j=1
αj(−i∂j), dom(H0) = [W 1,2(Rn)]N , (3.17)
where ∂j = ∂/∂xj, 1 6 j 6 n.
(iii) Next, consider the selfadjoint matrixvalued potential V = {V`,m}16`,m6Nsatisfying for some fixed ρ > 1, C ∈ (0,∞),
V ∈[L∞(Rn)
]N×N, V`,m(x) 6 C〈x〉−ρ for a.e. x ∈ Rn, 1 6 `,m 6 N . (3.18)
Under these assumptions on V , the massless Dirac operator H in [L2(Rn)]N isdefined via
H = H0 + V, dom(H) = dom(H0) = [W1,2(Rn)]N . (3.19)
Here we employed the shorthand notation
[L2(Rn)]N = L2(Rn;CN ), [W 1,2(Rn)]N = W 1,2(Rn;CN ), etc. (3.20)
Then H0 and H are selfadjoint in [L2(Rn)]N , with essential spectrum covering the
entire real line,
σess(H) = σess(H0) = σ(H0) = R, (3.21)a consequence of relative compactness of V with respect to H0. In addition,
σac(H0) = R, σp(H0) = σsc(H0) = ∅. (3.22)
With the exception of the comment following (3.25) and one more in connectionwith spectral shift functions in Remark 3.5, we will now drop the selfadjointness

12 F. GESZTESY AND R. NICHOLS
hypothesis on the N×N matrix V and still define a closed operator H in [L2(Rn)]Nas in (3.19).
Turning to the the Green’s matrix of the massless free Dirac operator H0 weassume
z ∈ C+, x, y ∈ Rn, x 6= y, n ∈ N, n > 2, (3.23)and compute for the Green’s function G0(z; · , · ) of H0,
G0(z;x, y) := (H0 − zI)−1(x, y)
= i4−1(2π)(2−n)/2x− y2−nz [zx− y](n−2)/2H(1)(n−2)/2(zx− y)IN (3.24)
− 4−1(2π)(2−n)/2x− y1−n[zx− y]n/2H(1)n/2(zx− y)α ·(x− y)x− y
.
The Green’s function G0(z; · , · ) of H0 continuously extends to z ∈ C+. In addition,in the massless case m = 0, the limit z → 0 exists,
limz→0,
z∈C+\{0}
G0(z;x, y) := G0(0;x, y)
= i2−1π−n/2Γ(n/2)α · (x− y)x− yn
, x, y ∈ Rn, x 6= y, n ∈ N, n > 2,(3.25)
and no blow up occurs for all n ∈ N, n > 2. This observation is consistent withthe sufficient condition for the Dirac operator H = H0 + V (in dimensions n ∈ N,n > 2), with V an appropriate selfadjoint N ×N matrixvalued potential, havingno eigenvalues, as derived in [19, Theorems 2.1, 2.3].
Returning to our analysis of the resolvent of H0, the asymptotic behavior (3.9)–(3.11) implies for some cn ∈ (0,∞),‖G0(0;x, y)‖B(CN ) 6 cnx− y1−n, x, y ∈ Rn, x 6= y, n ∈ N, n > 2, (3.26)
and for given R > 1,
‖G0(z;x, y)‖B(CN ) 6 cn,R(z)e−Im(z)x−y
x− y1−n, x− y 6 1, x 6= y,1, 1 6 x− y 6 R,x− y(1−n)/2, x− y > R,
z ∈ C+, x, y ∈ Rn, x 6= y, n ∈ N, n > 2, (3.27)
for some cn,R( · ) ∈ (0,∞) continuous and locally bounded on C+.For future purposes we now rewrite G0(z; · , · ) as follows:
G0(z;x, y) = i4−1(2π)(2−n)/2x− y2−nz [zx− y](n−2)/2H(1)(n−2)/2(zx− y)IN
− 4−1(2π)(2−n)/2x− y1−n[zx− y]n/2H(1)n/2(zx− y)α ·(x− y)x− y
= x− y1−nfn(z, x− y), (3.28)z ∈ C+, x, y ∈ Rn, x 6= y, n ∈ N, n > 2,
where fn is continuous and locally bounded on C+ × Rn, in addition,
‖fn(z, x)‖B(CN ) 6 cn(z)e−Im(z)x{
1, 0 6 x 6 1,x(n−1)/2, x > 1,
z ∈ C+, x, y ∈ Rn,(3.29)

ON A CLASS OF INTEGRAL OPERATORS 13
for some constant cn( · ) ∈ (0,∞) continuous and locally bounded on C+. In particular, decomposing G0(z; · , · ) into
G0(z;x, y) = G0(z;x, y)χ[0,1](x− y) +G0(z;x, y)χ[1,∞)(x− y):= G0,(z;x− y), (3.30)
z ∈ C+, x, y ∈ Rn, x 6= y, n ∈ N, n > 2,one verifies that
G0,>(z;x− y)j,k 6
{Cnx− y−(n−1), z = 0,Cn(z)x− y−(n−1)/2, z ∈ C+,
x, y ∈ Rn, x− y > 1, 1 6 j, k 6 N,(3.31)
for some constants Cn, Cn( · ) ∈ (0,∞), in particular,G0,>(z; · ) ∈ [L∞(Rn)]N×N , z ∈ C+, (3.32)
and that G0,>( · ; · ) is continuous on C+ × Rn.Starting our analysis of integral operators connected to the resolvent of H0 we
first note that Theorem 2.2 implies the following fact.
Theorem 3.3. Let n ∈ N, n > 2. Then the integral operator R0(δ) in [L2(Rn)]Nwith integral kernel R0(δ; · , · ) bounded entrywise byR0(δ; · , · )j,k 6 C〈 · 〉−δG0(0; · , · )j,k〈 · 〉−δ, δ > 1/2, 1 6 j, k 6 N, (3.33)
for some C ∈ (0,∞), is bounded,R0(δ) ∈ B
([L2(Rn)]N
). (3.34)
In a similar fashion, the integral operator R0(z, δ) in [L2(Rn)]N , with integral kernel
R0(z, δ; · , · ) bounded entrywise byR0(z, δ; · , · )j,k 6 C〈 · 〉−δG0(z; · , · )j,k〈 · 〉−δ,
δ > (n+ 1)//4, z ∈ C+, 1 6 j, k 6 N,(3.35)
for some C ∈ (0,∞), is bounded,R0(z, δ) ∈ B
([L2(Rn)]N
), z ∈ C+. (3.36)
Proof. The inclusion (3.34) is an immediate consequence of (3.25) and hence theestimate G0(0;x, y)j,k 6 Cx − y1−n, x, y ∈ Rn, x 6= y, 1 6 j, k 6 N , Theorem2.2, choosing c = d = 1/2 in (2.2), and an application of Theorem A.2 (i) andRemark A.3.
To prove the inclusion (3.36) we employ the estimates (3.9)–(3.11) (cf. also(3.27)) to obtain
G0(z;x, y)j,k 6 C(z)x− y1−nχ[0,1](x− y)
+D(z)x− y(1−n)/2χ[1,∞)(x− y), (3.37)z ∈ C+, x, y ∈ Rn, x 6= y, 1 6 j, k 6 N,
for some C,D(z) ∈ (0,∞), and apply Theorems 2.2 and A.2 (i) (cf. also RemarkA.3) to both terms on the righthand sides of (3.37). The part 0 6 x−y 6 1 leadsto δ > 1/2, whereas the part x− y > 1 yields δ > (n+ 1)/4, implying (3.36). �
Combining Theorems 2.1, 2.2, 2.8, and 3.1 then yields the second principal resultof this section, an application to massless Diractype operators.

14 F. GESZTESY AND R. NICHOLS
Theorem 3.4. Let n ∈ N, n > 2. Then the integral operator R0(δ) in [L2(Rn)]Nwith integral kernel R0(δ; · , · ) permitting the entrywise bound
R0(δ; · , · )j,k 6 C〈 · 〉−δG0(0; · , · )j,k〈 · 〉−δ, δ > 1/2, 1 6 j, k 6 N, (3.38)
for some C ∈ (0,∞), satisfies
R0(δ) ∈ Bp([L2(Rn)]N
), p > n. (3.39)
In a similar fashion, the integral operator R0(z, δ) in [L2(Rn)]N with integral kernel
R0(z, δ; · , · ) permitting the entrywise bound
R0(z, δ; · , · )j,k 6 C〈 · 〉−δG0(z; · , · )j,k〈 · 〉−δ,z ∈ C+, δ > (n+ 1)/4, 1 6 j, k 6 N,
(3.40)
for some C ∈ (0,∞), satisfies
R0(z, δ) ∈ Bp([L2(Rn)]N
), p > n, z ∈ C+. (3.41)
Proof. We will apply the fact (A.5).The inclusion (3.39) is immediate from (3.25) (employing the elementary esti
mate G0(0;x, y)j,k 6 Cx − y1−n, x, y ∈ Rn, x 6= y, 1 6 j, k 6 N) and Theorem3.1 (with γ = 1/2).
To prove the inclusion (3.41) we again employ the estimate (3.37). An applicationof Theorem 3.1 to both terms in (3.37), then yields for the part where 0 6 x−y 6 1that γ = 1/2 and hence δ > 1/2 and p > n. Similarly, for the part where x−y > 1one infers γ = (n + 1)/4 and hence δ > (n + 1)/4 and p > 2n/(n + 1), p > 2, andthus one concludes δ > (n+ 1)/4 and p > n. �
Remark 3.5. To put Theorem 3.4 a bit into perspective we note that inclusionsof the type (3.39), even in the far weaker situation with Bp
([L2(Rn)]N
)replaced
by B([L2(Rn)]N
), imply a global limiting absorption principle with strong spectral
implications (such as, the absence of any singular spectrum) for the underlyingDiractype operators, H0 and H = H0+c V , for sufficiently small coupling constantsc ∈ C. (For details in this limiting absorption context context we refer to [4], [5],[25, Sects. XIII.7, XIII.8], [30, Ch. 4], [31, Chs. 1, 2, 6] and the detailed bibliographycited therein). The actual Bp
([L2(Rn)]N
)result in Theorem 3.4 permits one to go
a step further and derive continuity properties of the spectral shift function (cf.,e.g., [30, Ch. 8], [31, Ch. 9]) between the pair of selfadjoint operators (H,H0) (herewe again assume the N × N matrixvalued potential V to be selfadjoint), whichin turn permits a discussion of the Witten index of class of nonFredholm modeloperators as discussed in [3]–[9], [12], [24], with additional material in preparation.
�
We conclude this section by noting once more that massless Dirac operators,particularly, in two dimensions, are known to be of relevance in applications tographene. This fact, and particularly the prominent role massless Diractype operators play in connection with the Witten index of certain classes of nonFredholmoperators, explains our interest in them.
Appendix A. Some Remarks on Block Matrix Operators
In this appendix we collect some useful (and wellknown) material on pointwise domination of linear operators in connection with boundedness, compactness,

ON A CLASS OF INTEGRAL OPERATORS 15
and the Hilbert–Schmidt property, with particular emphasis on the block matrixoperator situation (required in the context of Diractype operators).
Definition A.1. Let (M ;M;µ) be a σfinite, separable measure space, µ a nonnegative, measure with 0 < µ(M) 6∞, and consider the linear operators A,B definedon L2(M ; dµ). Then B pointwise dominates A
if for all f ∈ L2(M ; dµ), (Af)( · ) 6 (Bf )( · ) µa.e. on M. (A.1)
For a linear block operator matrix T = {Tj,k}16j,k6N , N ∈ N, in the Hilbertspace [L2(M ; dµ)]N (where [L2(M ; dµ)]N = L2(M ; dµ;CN ), we recall that T ∈B2([L2(M ; dµ)]N
)if and only if Tj,k ∈ B2
(L2(M ; dµ)
), 1 6 j, k 6 N . Moreover, we
recall that (cf. e.g., [2, Theorem 11.3.6])
‖T‖2B2(L2(M ;dµ)N ) =ˆM×M
dµ(x) dµ(y) ‖T (x, y)‖2B2(CN )
=
ˆM×M
dµ(x) dµ(y)
N∑j,k=1
Tj,k(x, y)2
=
N∑j,k=1
ˆM×M
dµ(x) dµ(y) Tj,k(x, y)2
=
N∑j,k=1
‖Tj,k‖2B2(L2(M ;dµ)), (A.2)
where, in obvious notation, T ( · , · ) denotes the N ×N matrixvalued integral kernel of T in [L2(M ; dµ)]N , and Tj,k( · , · ) represents the integral kernel of Tj,k inL2(M ; dµ), , 1 6 j, k 6 N .
In addition, employing the fact that for any N ×N matrix D ∈ CN×N ,‖D‖B(CN ) 6 ‖D‖B2(CN ) 6 N
1/2‖D‖B(CN ), (A.3)one also obtains
‖T‖2B2(L2(M ;dµ)N ) 6 NˆM×M
dµ(x) dµ(y) ‖T (x, y)‖2B(CN ). (A.4)
More generally, for H a complex separable Hilbert space and T = {Tj,k}16j,k6N ,N ∈ N, a block operator matrix in HN , one confirms thatT ∈ B
(HN )
(resp., T ∈ Bp
(HN
), p ∈ [1,∞) ∪ {∞}
)if and only if (A.5)
for each 1 6 j, k 6 N , Tj,k ∈ B(HN )
(resp., Tj,k ∈ Bp
(HN
), p ∈ [1,∞) ∪ {∞}
).
In other words, for membership of T in B(HN ) or Bp
(HN
), p ∈ [1,∞) ∪ {∞}, it
suffices to focus on each of its matrix elements Tj,k, 1 6 j, k 6 N . (For necessityof the last line in (A.5) it suffices to multiply T from the left and right by N ×Ndiagonal matrices with IH on the jth and kth position, respectively, and zerosotherwise, to isolate Tj,k and appeal to the ideal property. For sufficiency, it sufficesto write T as a sum of N2 terms with Tj,k at the j, kth position and zeros otherwise.)
The next result is useful in connection with Section 3.
Theorem A.2. Let N ∈ N and suppose that T1, T2 are linear N×N block operatormatrices defined on [L2(M ; dµ)]N , such that for each 1 6 j, k 6 N , T2,j,k pointwise

16 F. GESZTESY AND R. NICHOLS
dominates T1,j,k. Then the following items (i)–(iii) hold:
(i) If T2 ∈ B([L2(M ; dµ)]N
)then T1 ∈ B
([L2(M ; dµ)]N
)and
‖T1‖B([L2(M ;dµ)]N ) 6 ‖T2‖B([L2(M ;dµ)]N ). (A.6)
(ii) If T2 ∈ B∞([L2(M ; dµ)]N
)then T1 ∈ B∞
([L2(M ; dµ)]N
)and
‖T1‖B([L2(M ;dµ)]N ) 6 ‖T2‖B([L2(M ;dµ)]N ). (A.7)
(iii) If T2 ∈ B2([L2(M ; dµ)]N
)then T1 ∈ B2
([L2(M ; dµ)]N
)and
‖T1‖B2([L2(M ;dµ)]N ) 6 ‖T2‖B2([L2(M ;dµ)]N ). (A.8)
Proof. For item (ii) we refer to [11] and [23] (see also [21]) combined with (A.5) aswe will not use it in this paper. While the proofs of items (i) and (iii) are obviouslywellknown, we briefly recall them here as we will be using these facts in Section 3.Starting with item (i), we introduce the notation f = (f1, . . . , fN ) ∈ [L2(M ; dµ)]Nand f  = (f1, . . . , fN ) ∈ [L2(M ; dµ)]N and compute,
‖T1f‖2[L2(M ;dµ)]N =N∑j=1
‖(T1f)j‖2L2(M ;dµ) =N∑j=1
((T1f)j , (T1f)j)L2(M ;dµ)
=
N∑j=1
∣∣∣∣ N∑k,`=1
(T1,j,kfk, T1,j,`f`)L2(M ;dµ)
∣∣∣∣6
N∑j=1
N∑k,`=1
(T1,j,kfk, T1,j,`f`)L2(M ;dµ)
6N∑j=1
N∑k,`=1
(T1,j,kfk, T1,j,`f`)L2(M ;dµ)
6N∑j=1
N∑k,`=1
(T2,j,kfk, T2,j,`f`)L2(M ;dµ)
=
N∑j=1
((T2f )j , (T2f )j)L2(M ;dµ) = ‖T2f ‖2[L2(M ;dµ)]N
6 ‖T2‖2B(L2(M ;dµ)N )‖f ‖2[L2(M ;dµ)]N
= ‖T2‖2B(L2(M ;dµ)N )‖f‖2[L2(M ;dµ)]N , (A.9)
implying item (i). For item (iii) we recall from [27, Theorem 2.13] that T1,j,k ∈B2(L2(M ; dµ)
), 1 6 j, k 6 N , and ‖T1,j,k‖B2(L2(M ;dµ)) 6 ‖T2,j,k‖B2(L2(M ;dµ)), 1 6
j, k 6 N , and hence by (A.2),
‖T1‖2B2([L2(M ;dµ)]N ) =N∑
j,k=1
‖T1,j,k‖2B2(L2(M ;dµ)) 6N∑
j,k=1
‖T2,j,k‖2B2(L2(M ;dµ))
= ‖T2‖2B2([L2(M ;dµ)]N ).
(A.10)
�
Remark A.3. We note that the subordination assumption (Af)( · ) 6 (Bf )( · ) µa.e. on M , if A and B are integral operators in H with integral kernels A( · , · ) and

ON A CLASS OF INTEGRAL OPERATORS 17
B( · , · ), respectively, is implied by the condition A( · , · ) 6 B( · , · ) µ⊗ µa.e. onM ×M since
(Af)(x) =∣∣∣∣ˆM
dµ(y)A(x, y)f(y))
∣∣∣∣ 6 ˆM
dµ(y) A(x, y)f(y)
6ˆM
dµ(y)B(x, y)f(y) = (Bf )(x) for a.e. x ∈M . (A.11)
�
Next, we state the following result.
Lemma A.4. Let n ∈ N and suppose that K : Rn × Rn → [0,∞) satisfies
0 6 K(x, y) 6n∏j=1
Kj(xj , yj), x = (xj)nj=1, y = (yj)
nj=1 ∈ Rn, (A.12)
for functions Kj : R × R → [0,∞). If Kj( · , · ) is the kernel of a bounded integraloperator Kj ∈ B(L2(R)) for each 1 6 j 6 n, then K( · , · ) is the kernel of a boundedintegral operator K ∈ B(L2(Rn)), and
‖K‖B(L2(Rn)) 6n∏j=1
‖Kj‖B(L2(R)). (A.13)
Proof. We proceed by induction on n. The claim is evident in the case n = 1. Letn ∈ N, and suppose the claim is true for n− 1 ∈ N. In order to establish the claimfor n, we compute for f ∈ L2(Rn):ˆRndnx
∣∣∣∣ˆRndny K(x, y)f(y)
∣∣∣∣26ˆRdx1 · · ·
ˆRdxn
(ˆRdy1 · · ·
ˆRdyn
n∏j=1
K(xj , yj)f(y1, . . . , yn−1, yn))2
6 ‖K1‖2B(L2(R)) · · · ‖Kn−1‖2B(L2(R))
×ˆRdxn
∥∥∥∥ˆRdynKn(xn, yn)f(y1, . . . , yn−1, yn)
∥∥∥∥2L2(Rn−1;dy1···dyn−1)
, (A.14)
andˆRdxn
∥∥∥∥ˆRdynKn(xn, yn)f(y1, . . . , yn)
∥∥∥∥2L2(Rn−1;dy1···dyn−1)
=
ˆRdxn
[ˆRdy1 · · ·
ˆRdyn−1
(ˆRdynKn(xn, yn)f(y1, . . . , yn−1, yn)
)2]=
ˆRdy1 · · ·
ˆRdyn−1
[ˆRdxn
(ˆRdynKn(xn, yn)f(y1, . . . , yn−1, yn)
)2]6 ‖Kn‖2B(L2(R))
ˆRdy1 · · ·
ˆRdyn−1
(ˆRdyn f(y1, . . . , yn−1, yn)2
)= ‖Kn‖2B(L2(R))
ˆRndy1 · · · dyn f(y1, . . . , yn−1, yn)2
= ‖Kn‖2B(L2(R))‖f‖2L2(Rn). (A.15)

18 F. GESZTESY AND R. NICHOLS
To obtain the inequality in (A.15), we used the boundedness property of Kn in theformˆ
Rdxn
(ˆRdynKn(xn, yn)f(y1, . . . , yn−1, yn)
)2(A.16)
6 ‖Kn‖2B(L2(R))ˆRdyn f(y1, . . . , yn−1, yn)2 for a.e. (yj)n−1j=1 ∈ R
n−1. (A.17)
The claim and the estimate in (A.13) now follow upon combining (A.14) and (A.15).�
We conclude with one more fact from [17, Theorem 319]:
Lemma A.5. Let p ∈ (1,∞). If K : R × R → [0,∞) is homogeneous of degree(−1) and the (necessarily identical ) quantitiesˆ ∞
0
dsK(s, 1) s−1/p′
and
ˆ ∞0
dtK(1, t) t−1/p (A.18)
are equal to some number C ∈ (0,∞), then the integral operator K with kernelK( · , · ) belongs to B(Lp((0,∞))) and
‖K‖B(Lp((0,∞))) 6 C. (A.19)
Acknowledgments. We are indebted to Alan Carey, Jens Kaad, Galina Levitina,Denis Potapov, Fedor Sukochev, and Dima Zanin for helpful discussions and to thereferee for a very careful reading of our manuscript.
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20 F. GESZTESY AND R. NICHOLS
Department of Mathematics, Baylor University, One Bear Place #97328, Waco, TX
767987328, USA
Email address: Fritz [email protected]: http://www.baylor.edu/math/index.php?id=935340
Mathematics Department, The University of Tennessee at Chattanooga, 415 EMCSBuilding, Dept. 6956, 615 McCallie Ave, Chattanooga, TN 37403, USA
Email address: [email protected]
URL: http://www.utc.edu/faculty/rogernichols/index.php
1. Introduction2. Some Background Material3. Interpolation and Trace Ideal Properties of a Class of Integral OperatorsAppendix A. Some Remarks on Block Matrix OperatorsReferences