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 T γ,δ in L 2 (R n ), n N, n > 2, with integral kernels T γ,δ ( · , · ) bounded (Lebesgue) a.e. by |T γ,δ (x, y)| 6 Chxi -δ |x - y| 2γ-n hyi -δ , x, y R n ,x 6= y, for fixed C (0, ), 0 < 2γ<n, δ>γ, and prove that T γ,δ ∈Bp ( L 2 (R n ) ) for p > n/(2γ),p > 2. (Here hxi := (1 + |x| 2 ) 1/2 , x R n , and Bp abbreviates the p -based trace ideal.) These integral operators (and their matrix-valued analogs) naturally arise in the study of multi-dimensional Schr¨odinger and Dirac-type operators and we describe an application to the case of massless Dirac-type operators. 1. Introduction The principal aim of this paper is to derive trace ideal properties of a class of (matrix-valued) integral operators that naturally arise in the context of multi- dimensional Schr¨ odinger and Dirac-type operators. More precisely, we will focus on integral operators T γ,δ in L 2 (R n ), n N, n > 2, which in the scalar context are associated with integral kernels T γ,δ ( · , · ) that are bounded (Lebesgue) a.e. by |T γ,δ (x, y)| 6 Chxi -δ |x - y| 2γ-n hyi -δ , x, y R n ,x 6= y, (1.1) for fixed C (0, ), 0 < 2γ<n, δ>γ . We then prove in Theorem 3.1 that T γ,δ ∈B p ( L 2 (R n ) ) for p > n/(2γ ),p > 2. (1.2) This result is then applied to prove Theorem 3.4 which derives a uniform trace ideal norm bound with respect to the spectral parameter on the resolvent of mass- less Dirac-type operators if the Dirac-type resolvent is viewed as a map between suitable weighted L 2 (R n ) spaces. Such estimates are well-known to imply limiting absorption principles for the Dirac-type operator in question which, in turn, have strong spectral implications (such as the absence of any singular continuous spec- trum, etc.). Moreover, the fact that trace ideal bounds are involved can now be used 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 Dirac-type 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

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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〉−δ|x− y|2γ−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 + |x|2)1/2, x ∈ Rn, and Bp abbreviates the `p-based traceideal.) These integral operators (and their matrix-valued analogs) naturallyarise in the study of multi-dimensional Schrödinger and Dirac-type operators

    and we describe an application to the case of massless Dirac-type operators.

    1. Introduction

    The principal aim of this paper is to derive trace ideal properties of a classof (matrix-valued) integral operators that naturally arise in the context of multi-dimensional Schrödinger and Dirac-type 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〉−δ|x− y|2γ−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 mass-less Dirac-type operators if the Dirac-type resolvent is viewed as a map betweensuitable weighted L2(Rn) spaces. Such estimates are well-known to imply limitingabsorption principles for the Dirac-type operator in question which, in turn, havestrong spectral implications (such as the absence of any singular continuous spec-trum, 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 Dirac-type 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 non-Fredholm op-erators as discussed, for instance, in [3]–[9], [12], [24] (see also Remark 3.5). We notethat two-dimensional massless Dirac-type 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 right-hand 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 well-known 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 Dirac-type 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 Dirac-type 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 self-adjoint 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 cor-responding `p-based trace ideals will be denoted by Bp(H), their norms are abbre-viated 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 + |x|2)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 oper-ators, 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− y|λ

    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 Niren-berg and Walker [22] also to be employed in the proof of Theorem 3.1. For conve-nience 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|−c|x− y|(c+d)−n|y|−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

    |x|c|x− y|n−c−d|y|d, x ∈ Rn\{0}, (2.9)

    as |x| → ∞, one writesˆBn(0;1)

    dny

    |x|c|x− y|n−c−d|y|d=

    1

    |x|n−d

    ˆBn(0;1)

    dny∣∣∣∣ x|x| − y|x|∣∣∣∣n−c−d|y|d

    ,

    x ∈ Rn\{0}. (2.10)

    If x, y ∈ Rn with |x| > 2 and |y| 6 1, then the elementary estimates1

    26 1− 1

    |x|6

    ∣∣∣∣ x|x| − y|x|∣∣∣∣ 6 1 + 1|x| 6 32 (2.11)

  • 4 F. GESZTESY AND R. NICHOLS

    imply

    C1 6

    ∣∣∣∣ x|x| − y|x|∣∣∣∣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

    |y|d. (2.13)

    By Lebesgue’s dominated convergence theorem,

    lim|x|→∞

    ˆBn(0;1)

    dny∣∣∣∣ x|x| − y|x|∣∣∣∣n−c−d|y|d

    =

    ˆBn(0;1)

    dny lim|x|→∞

    1∣∣∣∣ x|x| − y|x|∣∣∣∣n−c−d|y|d

    =

    ˆBn(0;1)

    dny

    |y|d=: I1,n,d, (2.14)

    since (2.11) implies

    lim|x|→∞

    ∣∣∣∣ x|x| − y|x|∣∣∣∣ = 1, y ∈ Rn. (2.15)

    Therefore, by (2.10) and (2.14),ˆBn(0;1)

    dny

    |x|c|x− y|n−c−d|y|d∼ I1,n,d ·

    1

    |x|n−das |x| → ∞, (2.16)

    and, similarly,ˆBn(0;1)

    dnx

    |x|c|x− y|n−c−d|y|d∼ I1,n,c ·

    1

    |y|n−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

    |x||x− y|

    6 1 +|y||x− y|

    , x, y ∈ Rn, x 6= y, (2.18)

    implies

    Kc,d(x, y) =

    (|x||x− y|

    )−c1

    |x− y|n−d|y|d6M−c

    (1 +

    |y|−c

    |x− y|−c

    )1

    |x− y|n−d|y|d

    =M−c

    |x− y|n−d|y|d+

    M−c|x− y|n−(c+d)|y|c+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 right-hand 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 arithmetic-geometric 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/n|xj − yj |1−(c+d)/n|yj |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

    |s|c/n|s− t|1−(c+d)/n|t|d/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− 1|1−(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− y|β−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 well-known Schur criteria or Schur tests to which we briefly turn next forthe sake of completeness.

    The following well-known version of the Schur test (cf., e.g., [16, Theorem 5.2])provides a sufficient condition for boundedness between L2-spaces in terms of point-wise 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 σ-finitemeasure 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)

    ]1/2|f(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 Lp-based 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 σ-finite 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 µ-a.e. x ∈ X and ν-a.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 Lp-space, they do not yield information aboutpossible compactness or trace ideal properties of the integral operator. (For com-pactness 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 σ-finite 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 em-ploying 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〉−δ|x−y|2γ−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 n-dimensional massless Dirac-type 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〉−δ|x− y|2γ−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− y|ζ〈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 of|Tγ,δ( · ; · , · )|,

    |Tγ,δ(ζ;x, y)|2 6 〈x〉−2δ−Re(ζ)|x− y|2Re(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 Bp-ideals 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 multi-dimensional Schrödinger and Dirac-typeoperators 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 non-interacting)Schrödinger and Dirac-type operators, a natural first step in studying Schrödingerand Dirac-type operators with nontrivial interaction terms (i.e., potentials). Em-ploying, for instance, [1, p. 360, 364], one obtains

    H(1)0 (ζ) =

    ζ→0(2i/π)ln(ζ) +O

    (|ln(ζ)||ζ|2

    ), (3.9)

    H(1)ν (ζ) =ζ→0−(i/π)2νΓ(ν)ζ−ν +

    {O(|ζ|min(ν,−ν+2)

    ), ν /∈ N,

    O(|ln(ζ)||ζ|ν

    )+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−1/2|x− y|

    )(2−n)/2H

    (1)(n−2)/2

    (z1/2|x− y|

    ), n > 2, z ∈ C\{0},

    1

    (n− 2)ωn−1|x− y|2−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 off-diagonal for n > 3,

    limz→0

    g0(z;x, y) = g0(0;x, y) =1

    (n− 2)ωn−1|x− y|2−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(z1/2|x− y|/2

    )[1 +O

    (z|x− y|2

    )]+

    1

    2πψ(1)

    +O(|z||x− y|2

    ), 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 prepa-rations. To rigorously define the free massless n-dimensional Dirac operators to bestudied in the sequel, we now introduce the following set of basic hypotheses as-sumed 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 anti-commuting 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 self-adjoint matrix-valued 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 short-hand 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 self-adjoint 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 self-adjointness

  • 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)/2|x− y|2−nz [z|x− y|](n−2)/2H(1)(n−2)/2(z|x− y|)IN (3.24)

    − 4−1(2π)(2−n)/2|x− y|1−n[z|x− y|]n/2H(1)n/2(z|x− 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− y|n

    , 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 self-adjoint N ×N matrix-valued 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 cn|x− y|1−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− y|1−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)/2|x− y|2−nz [z|x− y|](n−2)/2H(1)(n−2)/2(z|x− y|)IN

    − 4−1(2π)(2−n)/2|x− y|1−n[z|x− y|]n/2H(1)n/2(z|x− y|)α ·(x− y)|x− y|

    = |x− y|1−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 par-ticular, 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

    {Cn|x− 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 by|R0(δ; · , · )j,k| 6 C〈 · 〉−δ|G0(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 by|R0(z, δ; · , · )j,k| 6 C〈 · 〉−δ|G0(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 C|x − y|1−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− y|1−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 right-hand 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 Dirac-type 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〈 · 〉−δ|G0(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〈 · 〉−δ|G0(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 C|x − y|1−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 underlyingDirac-type 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 self-adjoint operators (H,H0) (herewe again assume the N × N matrix-valued potential V to be self-adjoint), whichin turn permits a discussion of the Witten index of class of non-Fredholm 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 Dirac-type oper-ators play in connection with the Witten index of certain classes of non-Fredholmoperators, explains our interest in them.

    Appendix A. Some Remarks on Block Matrix Operators

    In this appendix we collect some useful (and well-known) material on point-wise 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 Dirac-type operators).

    Definition A.1. Let (M ;M;µ) be a σ-finite, separable measure space, µ a nonneg-ative, 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 (B|f |)( · ) µ-a.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 matrix-valued integral ker-nel 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 obviouslywell-known, 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,k|fk|, T2,j,`|f`|)L2(M ;dµ)

    =

    N∑j=1

    ((T2|f |)j , (T2|f |)j)L2(M ;dµ) = ‖T2|f |‖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 (B|f |)( · ) µ-a.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⊗ µ-a.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)| = (B|f |)(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))

    ׈Rdxn

    ∥∥∥∥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

    76798-7328, 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/roger-nichols/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