Schatten-von Neumann classes of integral operators

In this work we establish sharp kernel conditions ensuring that the corresponding integral operators belong to Schatten-von Neumann classes. The conditions are given in terms of the spectral properties of operators acting on the kernel. As applications we establish several criteria in terms of different types of differential operators and their spectral asymptotics in different settings: compact manifolds, operators on lattices, domains in ${\mathbb R}^n$ of finite measure, and conditions for operators on ${\mathbb R}^n$ given in terms of anharmonic oscillators. We also give examples in the settings of compact sub-Riemannian manifolds, contact manifolds, strictly pseudo-convex CR manifolds, and (sub-)Laplacians on compact Lie groups.


Introduction
Let (Ω j , M j , µ j ) (j = 1, 2) be measure spaces respectively endowed with a σ-finite measure µ j on a σ-algebra M j of subsets of Ω j . We denote L 2 (µ j ) := L 2 (Ω j , µ j ) the complex Hilbert space of square integrable functions on Ω j . In this paper we give sharp sufficient conditions on integral kernels K = K(x, y) in order to ensure that the corresponding integral operators T f (x) = Ω 1 K(x, y)f (y)dµ 1 (y) (1.1) from L 2 (µ 1 ) into L 2 (µ 2 ) belong to different Schatten-von Neumann classes and in particular to the trace class when (Ω 1 , M 1 , µ 1 ) = (Ω 2 , M 2 , µ 2 ). When Ω = Ω 1 = Ω 2 possesses a Borel topological structure a trace formula in terms of the diagonal of the kernel can be deduced. If additionally Ω has a smooth manifold structure some sharp sufficient conditions on integral kernels K(x, y) for Schatten-von Neumann classes can be formulated in terms of the regularity of a certain order in either x or y, or both, and in terms of decay conditions at infinity. To briefly explain the approach of this paper, we can summarise it as follows: If we know spectral properties of an operator E and we known how it acts on the integral kernel of an integral operator T , we can draw conclusions about the spectral properties of T . More specifically, let (E 2 ) x and (E 1 ) y be operators acting on x and y variables, respectively, and suppose we know that (E 2 ) x (E 1 ) y K ∈ L 2 (µ 2 ⊗ µ 1 ), (1.2) or, more generally, belongs to mixed Lebesgue spaces. In this paper we give spectral conditions on E 2 and E 1 ensuring that the integral operator T in (1.1) belongs to the Schatten-von Neumann class S r (L 2 (µ 1 ), L 2 (µ 2 )), 0 < r < ∞. Such spectral conditions on E 2 and E 1 will be given: • in terms of membership of their inverses in Schatten-von Neumann classes; • or in terms of their spectral asymptotics, i.e. conditions that can be verified in practice. As an application we present several tests in terms of different types of operators in different settings. While the knowledge of the spectral asymptotics of an operator E implies also the Schatten-von Neumann properties for its inverse, the advantage of knowing the spectral asymptotics will be in a possibility to obtain refined properties of the decay of the singular numbers of the integral operator T .
The problem of finding such criteria on different kinds of domains is classical and has been much studied, e.g. the paper [1] by Birman and Solomyak is a good introduction to the subject. In particular, it is well known that the smoothness of the kernel is related to the behaviour of the singular numbers. In a recent paper [2] the authors have established sufficient sharp kernel conditions for operators on L 2 (M) for a compact manifold M.
In order to obtain criteria for general Schatten-von Neumann classes we will use the well-known method of factorisation introduced by Gohberg and Krein (cf. [3,Chap. 3, Section 10]). Some applications of this method have been developed by O'Brien in [4] and in [5] for applications to trace formulae of Schrödinger operators. The factorisation method has also been applied by the authors in [2] in order to obtain Sobolev regularity conditions for kernels on closed manifolds. The main result of [2] has been recently applied in [6] for the study of the distribution of eigenvalues for the double layer potential on flat curves and its relation with isoperimetrical inequalities.
There is an extensive literature on Schatten-von Neumann properties for specific integral operators. A characterisation of the membership in Schatten-von Neumann classes for Hardy operators has been established by Nowak in [7]. Lower bounds for the Schatten-von Neumann norms for Volterra integral operators have been obtained in [8], [9]. Schatten-von Neumann classes of pseudo-differential operators in the setting of the Weyl-Hörmander calculus have been considered in [10], [11], [12] and for anharmonic oscillators in [13]. Schatten-von Neumann classes on compact Lie groups and s-nuclear operators on L p spaces from the point of view of symbols have been respectively studied by the authors in [14] and [15].
In the subsequent part of the present paper we establish a characterisation of Schatten-von Neumann classes for operators from L 2 (µ 1 ) into L 2 (µ 2 ) for general integral operators on general measure spaces.
The role of the smoothness of the kernel in the investigation of Schatten-von Neumann properties can be illustrated with the following example. In his classical book [16,Prop 3.5, page 174] Mitsuo Sugiura gives a trace class criterion for integral operators on L 2 (T 1 ) with C 2 -kernels. More precisely, the theorem asserts that every kernel in C 2 (T 2 ) begets a trace class operator on L 2 (T 1 ): if K(θ, φ) is a C 2 -function on T 2 , then the integral operator L on L 2 (T 1 ) defined by (Lf )(θ) = The proof of this result relies on the connection between the absolute convergence of Fourier coefficients of the kernel and the trace class property (traceability) of the corresponding operator. This result has been extended by the authors under sharp smoothness assumptions to the setting of compact manifolds in [2]. The authors have weakened the known assumptions on the kernel for the operator to be trace class and for the trace formula (1.4) to hold. The present paper significantly extends the main results contained in [2] in different ways.
To formulate the notions more precisely, let H 1 , H 2 be complex Hilbert spaces and let T : H 1 → H 2 be a compact operator. Then T * T : H 1 → H 1 is compact, self-adjoint, and non-negative. Hence, we can define the absolute value of T by the equality |T | = (T * T ) 1 2 : H 1 → H 1 . Let (ψ k ) k be an orthonormal basis for H 1 consisting of eigenvectors of |T |, and let s k (T ) be the eigenvalue corresponding to the eigenvector ψ k , k = 1, 2, . . . . The numbers s 1 (T ) ≥ s 2 (T ) ≥ · · · ≥ s n (T ) ≥ · · · ≥ 0, are called the singular values of T : H 1 → H 2 . If 0 < p < ∞ and the sequence of singular values is ℓ p -summable, then T is said to belong to the Schatten-von Neumann class S p (H 1 , H 2 ). If 1 ≤ p < ∞, a norm is associated to S p (H 1 , H 2 ) by If 1 ≤ p < ∞ the class S p (H 1 , H 2 ) becomes a Banach space endowed with the norm T Sp . If p = ∞ we define S ∞ (H 1 , H 2 ) to be the class of compact operators from H 1 into H 2 endowed with the operator norm T S∞ := T op . In the case 0 < p < 1 the quantity T Sp only defines a quasinorm, and S p (H 1 , H 2 ) is also complete. If H 1 = H 2 = H we will simply write S p (H, H) = S p (H).
A relationship between the singular values s n (T ) and the eigenvalues λ n (T ) in the case H = H 1 = H 2 for an operator T ∈ S ∞ (H) was established by Hermann Weyl (cf. [19]): We will apply (1.6) for factorising our operators T in the form T = AB with A ∈ S q and B ∈ S p , and from this we deduce that T ∈ S r . We refer the reader to [17,Chapter 7] for more details on the Schatten-von Neumann classes for operators acting on different Hilbert spaces H 1 , H 2 . Standard references for the special case H 1 = H 2 are [3], [20], [18], [21]. If H = H 1 = H 2 and T ∈ S 1 (H) then T is said to be in the trace class S 1 . Let T : H → H be an operator in S 1 (H) and let (φ k ) k be any orthonormal basis for H. Then, the series ∞ k=1 (T φ k , φ k ) is absolutely convergent and the sum is independent of the choice of the orthonormal basis (φ k ) k . Thus, we can define the trace Tr(T ) of any linear operator T : H → H in S 1 by where {φ k : k = 1, 2, . . . } is any orthonormal basis for H. If the singular values are square-summable T is called a Hilbert-Schmidt operator. It is clear that every trace class operator is a Hilbert-Schmidt operator. A nice basic introduction to the study of the trace class is contained in the book [22] by Peter Lax.
We note that in view of (1.5) the condition K ∈ L 2 (Ω 2 × Ω 1 ) implies that T ∈ S p for all p ≥ 2. The situation for Schatten-von Neumann classes S p for p > 2 is indeed simpler and, in fact, similar to that of p = 2. In particular, B. Russo has proved in [23] that for µ 1 = µ 2 = µ on Ω: If 1 < p < 2, 1 p + 1 p ′ = 1 and K ∈ L 2 (Ω × Ω) then for the corresponding integral operator T one has where · p,q denotes the mixed-norm: and K * (x, y) = K(y, x). The condition K ∈ L 2 (Ω×Ω) in the above statement can be removed, see Goffeng [24]. Namely, even without such assumption, (1.9) still holds as long as its right hand side is finite. See also the discussion of such topics around [25,Theorem 2.3].
For p < 2, the situation is much more subtle, and the Schatten-von Neumann class S p (L 2 (µ 1 ), L 2 (µ 2 )) cannot be characterised as in the case p = 2 by a property analogous to the square integrability of integral kernels. This is a crucial fact that we now briefly describe. A classical result of Carleman [26]  Now, using this and considering the normal operator acting on L 2 (T 1 ) one obtains that the sequence (c k ) k forms a complete system of eigenvalues of this operator corresponding to the complete orthonormal system The system φ k is also complete for T * , T * φ k = c k φ k , so that the singular values of T are given by s k (T ) = |c k |, and hence by ( In other words, in contrast to the case of the class S 2 of Hilbert-Schmidt operators which is characterised by the square integrability of the kernel, Carleman's result shows that below the index p = 2 the class of kernels generating operators in the Schatten-von Neumann class S p cannot be characterised by criteria of the type |F (K(x, y))|dxdy < ∞, for any continuous function F since the kernel K(x, y) = κ(x − y) of the operator T in (1.12) satisfies any kind of integral condition of such form due to the boundedness of κ.
This example demonstrates the relevance of obtaining criteria for operators to belong to Schatten-von Neumann classes for p < 2 and, in particular, motivates the results in this paper. Among other things, we may also note that the continuity of the kernel (as in the above example) also does not guarantee that the operator would belong to any of the Schatten-von Neumann classes S p with p < 2. Therefore, it is natural to ask what regularity imposed on the kernel would guarantee such inclusions (for example, the C 2 condition in Sugiura's result mentioned earlier does imply the traceability on T 1 ). Thus, these questions will be the main interest of the present paper.
The main result for operators to belong to Schatten-von Neumann classes S p for 0 < p < 2, is given in Theorem 2.2. In this work we allow singularities in the kernel so that the formula (1.4) would need to be modified in order for the integral over the diagonal to make sense. In such case, in order to calculate the trace of an integral operator using a non-continuous kernel along the diagonal, one idea is to average it to obtain an integrable function. Such an averaging process has been introduced by Weidmann [27] in the Euclidean setting, and applied by Brislawn in [28], [29] for integral operators on L 2 (R n ) and on L 2 (Ω, M, µ), respectively, where Ω is a second countable topological space endowed with a σ-finite Borel measure. The corresponding extensions to the L p setting have been established in [30] and [31]. The L 2 regularity of such an averaging process is a consequence of the L 2 -boundedness of the martingale maximal function. Denoting by K(x, x) the pointwise values of this averaging process, Brislawn [29] proved the following formula for a trace class operator T on L 2 (µ), for the extension to L p see [30]: (1.13) In Section 2 we establish our criteria for Schatten-von Neumann classes on measure spaces, and the special case of the trace class is treated in Section 3. For this, we briefly recall the definition of the averaging process involved in formula (1.13). In Section 4 we present further criteria in terms of operators for which the distribution of eigenvalues is known in terms of the asymptotics of the eigenvalue counting functions. In Section 5 we give applications of the obtained conditions in different settings.
The authors would like to thank Alberto Parmeggiani for a discussion.

Schatten-von Neumann classes on L 2 -spaces
In this section we present our results in the setting of L 2 -spaces which is not restrictive in terms of the general theory of Hilbert spaces. Before stating our first result, we point out that a look at the proof of the trace formula (1.4) in [16,Prop 3.5] shows that statement can be already improved in the following way: ∂θ 2 + ∂ 2 ∂φ 2 be the Laplacian on T 2 . Let K(θ, φ) be a measurable function on T 2 and suppose that there exists an integer q > 1 such that ∆ q 2 K ∈ L 2 (T 1 × T 1 ). Then the integral operator L on L 2 (T 1 ) defined by is trace class and has the trace where K stands for the averaging process described in Section 3.
We observe that for K ∈ L 2 (µ 2 ⊗ µ 1 ), we have or we can also write ). In particular, this also means that K y defined by K y (x) = K(x, y) is well-defined for almost every y ∈ Ω 1 as a function in L 2 x (µ 2 ). For an operator E from L 2 (µ 1 ) into L 2 (µ 1 ) we will use the notation E x K(x, y) to emphasize that the operator E is acting on the x-variable. Analogously, we will also use the notation E y K(x, y) for an operator E from L 2 (µ 2 ) into L 2 (µ 2 ) acting on the y-variable.
In this paper we will only consider linear operators. We will now give our main criteria for Schatten-von Neumann classes where we note that we do not have to assume the operators E 1 , E 2 to be self-adjoint nor bounded, i.e. they are considered to be densely defined without further explanations. For a densely defined operator E on a Hilbert, it is well known that, E has a bounded inverse if and only if E is closed and bijective. Henceforth, an invertible operator is understood as an operator with bounded inverse. An unbounded operator is understood as a densely defined operator.
Proof of Theorem 2.2. (i) For the sake of simplicity, sometimes we will abbreviate the notation also in integrals by writing We now consider the operator A with kernel One can see that Since A ∈ S 2 (L 2 (µ 1 ), L 2 (µ 2 )) and using the fact that (E * 1 ) −1 ∈ S p 1 (L 2 (µ 1 )) if and only if E −1 1 ∈ S p 1 with equal norms, we obtain Moreover, for the estimation of the Schatten-von Neumann norm T Sr , according to (1.8) we obtain: , and this concludes the proof of (i).
(ii) Just consider E 2 in the proof of (i).
(iii) Just consider E 1 in the proof of (i).
Remark 2.4. (i) In the general setting of a separable Hilbert spaces H one can construct operators E satisfying the assumptions in the above theorem. Let 0 < p < ∞ and (s n ) n a sequence in ℓ p , with s n > 0 for all n. Let {φ n : n = 1, 2, . . . } be an orthonormal basis of H.

We define
Since Span({φ n : n = 1, 2, . . . }) ⊂ D, then D is dense in H. We define Eφ n = s −p n φ n . By using Cauchy-Schwarz inequality, we can see that E can be extended to D by We also note that since lim n s p n = 0, we have lim n s −p n = +∞ and the operator E is not bounded.
In more specific cases, we will consider more concrete operators for the applications.
(ii) We point out that a converse statement also holds for the multiplication property (1.6) (cf. [17,Theorem 7.9]): Let 0 < p, q, r < ∞ and T ∈ S r (H, Then there exist operators B ∈ S p (H, H 1 ) and A ∈ S q (H 1 , H 2 ) (with some Hilbert space H 1 ) for which T = AB; the operators A, B can be chosen such that T Sr(H, Remark 2.5. Under conditions of Theorem 2.2, in the proof of its Part (i) the main point was to obtain the factorisation . This factorisation has other consequences. For example, the combination of (2.6), the condition (1.9) and the multiplication property imply the following extension of Theorem 2.2 in the case µ 1 = µ 2 = µ on Ω, where we will denote by L q ′ (Ω, L q (Ω)) the space defined by the mixed norm (1.10), that is, by We also use the notation K * (x, y) := K(y, x).
Corollary 2.6. Let (Ω, M, µ) be a σ-finite measure space. Let T be a bounded integral operator on L 2 (Ω), defined by Let 1 < q ≤ 2 and 1 q + 1 q ′ = 1. Then the following holds: then T belongs to the Schatten-von Neumann classes S r (L 2 (Ω)) for all 0 < r < ∞ such that Moreover, Moreover, respectively one has respectively.
Remark 2.7. We note that following the remarks after (1.10) we do not need to assume in Corollary 2.6 that K ∈ L 2 (Ω × Ω). Consequently, compared with the sufficient condition (1.9) by Russo and with Theorem 2.2, the Schatten-von Neumann class index in Corollary 2.6 can be larger than 2. Indeed, compared with the argument in Remark 2.3, the condition on r in Corollary 2.6 becomes 0 < r < p 1 p 2 p 1 +p 2 in Part (i) and 0 < r < p in Part (ii), respectively. Therefore, even for Schatten-von Neumann classes S r with r > 2, Corollary 2.6 extends the sufficient condition (1.9) by Russo in the following sense: For an integral operator to belong to the Schatten-von Neumann classes S r with r > 2, the 'size' condition (1.9) can be relaxed if we know that the integral kernel of an integral operator has additional 'regularity' properties.

Trace class operators and their traces
In this section we consider the important case of the trace class operators. We start by deducing a corollary of Theorem 2.2 in this special case. In order to establish a formula for the trace we will require an additional topological structure on Ω. We will now briefly recall the averaging process which is required for the study of trace formulae for kernels with discontinuities along the diagonal. We start by defining the martingale maximal function. Let (Ω, M, µ) be a σ-finite measure space and let {M j } j be a sequence of sub-σ-algebras such that In order to define conditional expectations we assume that µ is σ-finite on each M j .
In order to obtain a generalisation of the Hardy-Littlewood maximal function we consider the particular case of martingales generated by a single M-measurable function f . The martingale maximal function is defined by This martingale can be defined, in particular, when the σ-algebra M is countably generated and it will allow to study the trace by mean of an averaging process on the diagonal of the kernel. However, this process is most effective for the computations in the case of a σ-algebra of Borel sets for a second countable topological space. Henceforth we will assume that Ω is a second countable topological space, M is the σ-algebra of Borel sets and µ is a σ-finite Borel measure. For our purposes in the study of the kernel the sequence of σ-algebras is constructed from a corresponding increasing sequence of partitions P j × P j of Ω × Ω. Now, for each (x, y) ∈ Ω × Ω there is a unique C j (x) × C j (y) ∈ P j × P j containing (x, y). Those sets C j (x) replace the cubes in R n in the definition of the classical Hardy-Littlewood maximal function. We refer to Doob [32] for more details on the martingale maximal function and its properties.
We denote by A (2) j the averaging operators on Ω × Ω: Let K ∈ L 1 loc (µ ⊗ µ), then the averaging A (2) j is defined µ ⊗ µ-almost everywhere (cf. [29]) by The averaging process will be applied to the kernels K(x, y) of our operators. As a consequence of the fundamental properties of the martingale maximal function it can be deduced that is defined almost everywhere and that it agrees with K(x, y) in the points of continuity. We observe that if K(x, y) is the integral kernel of a trace class operator, then K(x, y) is, in particular, square integrable on Ω × Ω. A classical example with a discontinuous kernel is the Volterra operator V on L 2 (I) where I = [0, 1]. Its kernel is given by By averaging on cubes one can see that K(x, x) = 1 2 for 0 < x < 1. However, it is well known that its singular values are s n = 2(π(2n + 1)) −1 , hence V is not a trace class operator.
In the sequel in this section, we can always assume that K ∈ L 2 (µ ⊗ µ) since it is not restrictive because the trace class is included in the Hilbert-Schmidt class, and the square integrability of the kernel is then a necessary condition.
(iii) Moreover, assume additionally that Ω is a second countable topological space and (Ω, M, µ) is a measure space endowed with a σ-finite Borel measure µ. Then under any of the assumptions (i) or (ii), the operator T is trace class on L 2 (µ) and its trace is given by In particular, if K is continuous on the diagonal one has Proof. By taking r = 1 in the corresponding assumptions in Corollary 2.6 one can deduce (i) and (ii). For (iii) the fact that T is trace class follows from the corresponding assumption (i) or (ii), and the trace formula comes from (1.13), with K given by (3.4). The last part, follows since K agrees with K in the points of continuity.
where λ j are the eigenvalues of the operator T counted with multiplicities.
(b) The additional assumption on Ω to be a second countable topological space is only required in order to obtain the additional formula (3.9). This requirement is enough general for the applications we will consider in this work.
(c) If Ω is a second topological space and K ∈ L 2 (µ⊗µ) we haveK(x, y) = K(x, y) for the points of continuity of K. Hence any continuous kernel on the diagonal provides an example where this limit can be obtained just as the pointwise value K(x, x). An example of relevance in spectral geometry is provided in Remark 5.2 (b) with the kernel of the double layer potential for a C 2 bounded region Ω in R 2 . Indeed, it is known from two dimensional potential theory that this kernel is continuous and

Conditions in terms of spectral asymptotics
The typical application of the results above may come from the observation that knowing the spectral asymptotics of E 1 and E 2 implies conclusions about the membership in Schatten-von Neumann classes for their inverses. However, in the case when the spectral asymptotics of operators E 1 and E 2 are available, the spectral conclusions for the integral operators can be sharpened further in terms of the decay rates of their singular numbers.
As further examples, in Section 5 we will consider different kinds of domains and operators to test the membership in the Schatten-von Neumann classes.
The following conditions are based on the knowledge of the behaviour of the eigenvalue counting function of the operators E 1 , E 2 . We recall that for a self-adjoint operator E with discrete spectrum {λ j } j its eigenvalue counting function is defined by N(λ) := #{j : λ j ≤ λ}, where λ j 's are counted with their respective multiplicities. The conditions that we will impose can be effectively verified as we will shown in the subsequent subsections.
Theorem 4.1. Let (Ω i , M i , µ i ) (i = 1, 2) be σ-finite measure spaces. For each i = 1, 2, let E i be an essentially self-adjoint operator on L 2 (µ i ) such that the spectrum of its closure consists of a sequence of discrete and strictly positive eigenvalues 0 < λ 1,i ≤ λ 2,i ≤ · · · , whose eigenvectors are a basis of L 2 (µ i ). Assume that for the eigenvalue counting function (4.1) Let K ∈ L 2 (µ 2 ⊗ µ 1 ) and let T be the integral operator from L 2 (µ 1 ) to L 2 (µ 2 ) defined by Then the following holds: , then T belongs to the Schatten-von Neumann class S r (L 2 (µ 1 ), L 2 (µ 2 )) for all 0 < r < ∞ such that and (2.1) holds.
Moreover, the sequence of singular values (s k (T )) k satisfies the following estimate for the rate of decay: p 2 ). (ii) Let E be an unbounded invertible operator on L 2 as above such that its spectrum satisfies (4.1) for some p > 0. If either E y K ∈ L 2 (µ 2 ⊗ µ 1 ) or E x K ∈ L 2 (µ 2 ⊗µ 1 ), then T belongs to the Schatten-von Neumann class S r (L 2 (µ 1 ), L 2 (µ 2 )) for all 0 < r < ∞ such that and respectively (2.2) or (2.3) holds.
Moreover, the sequence of singular values (s k (T )) k satisfies the following estimate for the rate of decay: Proof. (i) We note that the assumptions on N i for i = 1, 2, imply that and thus also Thus E −1 i is a compact operator and its singular values are s k (E −1 i ) = λ −1 k,i and E −1 i ∈ S q i (L 2 (µ i )) for all q i > p i . Now, for q i > p i the fact that T ∈ S r (L 2 (µ 1 ), L 2 (µ 2 )) can now be deduced from Theorem 2.2 and 1 2 In order to get the estimate for the rate of decay of the singular values we will use the following Fan's inequality (cf. [34], [18]) for the singular values of the composition of two compact operators: for all k, l ≥ 1. We will apply (4.4) to the factorisation T = E −1 2 A(E * 1 ) −1 obtained in the proof of Theorem 2.2. By using (4.4) with l + m − 1 instead of l we get Thus, with k = l = m we obtain Since (s k (T )) k is a non-increasing sequence, then s 3k (T ), s 3k−1 (T ) ≤ s 3k−2 (T ) and This concludes the proof of (i). The proof of (ii) follows in a similar way by considering the factorisation T = E −1 A so we can omit the details. If Ω 1 = Ω 2 = Ω and µ 1 = µ 2 = µ, the statement of Theorem 4.1 can be extended by using Corollary 2.6 instead of Theorem 2.2. More precisely, assume that E 1 and E 2 satisfy the assumptions of Theorem 4.1. Let 1 < q ≤ 2 and 1 q + 1 q ′ = 1. Then the following holds: (i) If (E 2 ) x (E 1 ) y K and ((E 2 ) x (E 1 ) y K) * ∈ L q ′ (Ω, L q (Ω)), then T belongs to the Schatten-von Neumann class S r (L 2 (Ω)) for all 0 < r < ∞ such that 1 and (2.8) holds.
Moreover, the sequence of singular values (s k (T )) k satisfies the following estimate for the rate of decay: (ii) Let E be an unbounded invertible operator on L 2 as above such that its spectrum satisfies (4.1) for some p > 0. If E x K, (E x K) * ∈ L q ′ (Ω, L q (Ω)) or E y K, (E y K) * ∈ L q ′ (Ω, L q (Ω)), then T belongs to the Schatten-von Neumann class S r (L 2 (Ω)) for all 0 < r < ∞ such that 1 and respectively (2.9) or (2.10) holds.
Moreover, the sequence of singular values (s k (T )) k satisfies the following estimate for the rate of decay:

Applications
In this section we will describe several example situations where one can apply the obtained results: • compact manifolds: taking E 1 , E 2 to be elliptic pseudo-differential operators one obtains conditions in terms of the regularity of the kernel; • lattices: here the regularity of the kernel becomes irrelevant; however, due to non-compactness the conditions are formulated in terms of the behaviour of the integral kernel at infinity; • R n : for domains which are not necessarily bounded but have finite Lebesgue measure in Section 5.2 we obtain conditions still only in terms of the regularity of the kernel; • R n : in general, due to non-boundedness the regularity of the kernel by itself is not sufficient to ensure the compactness of the operator, and the regularity assumptions should be combined with decay conditions at infinity. It is convenient to formulate such conditions in terms of the action of harmonic or anharmonic oscillators on the kernel; in particular, it shows that different combinations of regularity and decay may ensure the membership in the Schatten-von Neumann classes on R n ; • sub-Riemannian settings: here is may be natural to formulate the conditions in terms of the operators associated to the sub-Riemannian structure (such as the sub-Laplacian). In Section 5.6 we briefly discuss the implications for general compact sub-Riemannian manifolds, contact manifolds, strictly pseudo-convex CR manifolds, and (sub-)Laplacians on compact Lie groups. Thus, in the following subsections we consider several applications of Theorem 2.2, Corollary 3.1 and Theorem 4.1.
In the case when the operators act on the same space we also have natural extensions of the statements below by using mixed L q ′ (Ω, L q (Ω)) norms as in Corollary 2.6 and Remark 4.2 instead. For simplicity, we mostly restrict to the L 2 -case since the extensions to the L q ′ (Ω, L q (Ω)) setting are rather straightforward.

5.1.
Operators on closed manifolds. In this section we will consider the case of integral operators on a compact manifold without boundary.
Thus, let M be a smooth compact manifold without boundary of dimension n endowed with a volume element dx. We denote by Ψ ν +e (M) the class of positive elliptic pseudo-differential operators of order ν ∈ R, i.e. positive operators which in every coordinate chart are operators in Hörmander classes on R n with elliptic symbols in S ν 1,0 , see e.g. [35]. We note that for any positive elliptic operator P ∈ Ψ ν +e (M) the standard Sobolev space H µ (M) defined in local coordinates can be characterised as the space of all distributions f ∈ D ′ (M) such that (I + P ) µ ν f ∈ L 2 (M). Let now M 1 , M 2 be closed manifolds and P i ∈ Ψ ν i +e (M i ) (i = 1, 2) with ν i > 0. Consequently, the following mixed regularity Sobolev space H µ 2 ,µ 1 x,y (M 2 ×M 1 ) of mixed regularity µ 1 , µ 2 ≥ 0, defined by is independent of the choice of operators P 1 , P 2 . The relation between these mixed Sobolev spaces and the standard Sobolev spaces H µ (M 2 × M 1 ) on the manifold M 2 × M 1 is given by for all µ 1 , µ 2 ≥ 0. This can be readily seen by an extension of an argument in [2,Proposition 4.3] where this was shown to hold in the case of M 1 = M 2 . Then we have the following statement. We will write E i = (I + P i ) µ i ν i for i = 1, 2.
Corollary 5.1. Let M 1 , M 2 be closed manifolds of dimensions n 1 , n 2 , respectively, and let µ 1 , is in the Schatten-von Neumann classes S r (L 2 (M 1 ), L 2 (M 2 )) for Moreover, its singular numbers satisfy In particular, for M = M 1 = M 2 , n = n 1 = n 2 : then T is trace class on L 2 (M) and its trace is given by (3.8).
x C ℓ 2 y (M × M) for some even integers ℓ 1 , ℓ 2 ∈ 2N 0 such that ℓ 1 + ℓ 2 > n 2 , then T is trace class on L 2 (M) and its trace is given by Proof. In order to prove that T belongs to S r (L 2 (M 1 ), L 2 (M 2 )) with r satisfying (5.3) we first recall the following fact: if P ∈ Ψ ν +e (M) is a positive elliptic pseudodifferential operator of order ν > 0 on a closed manifold M of dimension n and 0 < p < ∞ then x ∈ S p 2 for p 2 > n 2 µ 2 , we have that T belongs to S r (L 2 (M 1 ), L 2 (M 2 )) for r > 0 as in (5.3). The rate of decay (5.4) is now a consequence of Theorem 4.1 and the spectral asymptotics for elliptic pseudo-differential operators on compact manifolds. Furthermore, Part (i) is obtained by letting r = 1 in (5.3), and Part (ii) follows from Part (i) and formula (3.9).
This corollary refines the results by the authors in [2] where the statement (5.3) was obtained in the case M 1 = M 2 . Now this and the refinement of the decay rate in (5.4) have been obtained as corollaries of Theorem 2.2 and Theorem 4.1.
Remark 5.2. (a) We can note that the index n 2 in Part (ii) of Corollary 5.1 is in general sharp. For example, for M = T n being the torus of even dimension n, there exist a function χ of class C n 2 such that the series of its Fourier coefficients diverges (see [36,Ch. VII] or [37]). By considering the convolution kernel K(x, y) = χ(x − y), the singular values of the operator T given by T f = f * χ agree with the absolute values of the Fourier coefficients of χ. Hence, T / ∈ S 1 (L 2 (T n )) but K ∈ C n 2 (M × M). Thus, we see that Part (ii) of Corollary 5.1 with ℓ 1 = 0 and ℓ 2 = n 2 is sharp. This, in turn, justifies the sharpness, in general, for all the results in this paper.
(b) An example that arises in spectral geometry is given by the two dimensional double layer potential. Let Ω be a C k bounded region in R 2 with k ≥ 2. Let E(x, y) = 1 π log 1 |x−y| , the double layer potential K : L 2 (∂Ω) → L 2 (∂Ω) is defined as the operator where ∂ν y denotes is the outer normal derivative. The kernel K(x, y) is continuous on ∂Ω×∂Ω, by studying its regularity depending on k as has been applied in [6] and using the results for closed manifolds, in this case for ∂Ω one can determine the rate of decay for the eigenvalues of the double layer potential from the corresponding membership of the double layer potential to a Schatten-von Neumann class. In particular, one can also deduce trace class properties. We refer to [6] for the details on this important example.

5.2.
Operators on domains with finite measure. In Section 5.1 we considered the case of compact domains. We now discuss the situation when the domains may be unbounded but still have finite measure.
x C ℓ 2 y (Ω×Ω) for some even integers ℓ 1 , ℓ 2 ∈ 2N 0 such that ℓ 1 +ℓ 2 > n 2 , then T is trace class on L 2 (Ω) and its trace is given by We note that Corollary 5.3 applies to domains that do not have to be bounded but have finite measure. If the measure of the domain is infinite the regularity of the kernel may not be enough and should be complemented by decay conditions at infinity. Such situations will be considered in Section 5.4 and Section 5.5.
Applications of Theorem 4.1 to obtain further refinements on the decay rate of singular numbers of T are possible, however, the spectral asymptotics required for its use for fractional Laplacians in Ω could in general depend on properties of the boundary and boundary conditions, and thus would require further assumptions.

Operators on lattices.
In this section we consider operators acting on functions on the integer lattice Z n . Compared to Section 5.1, here the decay conditions at infinity are important while the regularity of the kernel becomes irrelevant (as regularity of a pointwise defined function on a discrete lattice).
We note that as before, Part (i) of the following statement is a special case of Part (ii) with q = 2, but in the case of n = m, i.e. when the integral operator is acting on the same space. It will be useful to employ the operator (5.14) In cases when there are several variables, we will also write E α k for E α to emphasise that the operator is acting in the variable k.
Corollary 5.4. Let n, m ∈ N. Let K : Z n × Z m → C be a function and let T be the operator, bounded from ℓ 2 (Z m ) to ℓ 2 (Z n ), defined by Then we have the following properties.
(i) Assume that for some α, β ≥ 0 we have Then T ∈ S r (ℓ 2 (Z m ), ℓ 2 (Z n )) for all 0 < r < ∞ such that The sequence of singular values (s j (T )) j satisfies the following estimate for the rate of decay: then the operator T is trace class on ℓ 2 (Z n ) and its trace is given by 19) where λ j are the eigenvalues of the operator T counted with multiplicities. (ii) In the case m = n, let 1 < q ≤ 2 and 1 q + 1 q ′ = 1. Assume that for some α, β ≥ 0 we have Then T ∈ S r (ℓ 2 (Z n )) for all 0 < r < ∞ such that Moreover, the sequence of singular values (s j (T )) j satisfies the following estimate for the rate of decay: In particular, if α, β ≥ 0 are such that α + β > n q , then T is trace class on ℓ 2 (Z n ) and its trace is given by (5.19).
Proof. Part (i). We observe that the assumption (5.15) of Corollary 5.4 can be formulated as E α k E β l K ∈ ℓ 2 (Z n × Z m ). (5.24) In order to apply Theorem 4.1 we first note that the Kronecker's delta δ k is an eigenfunction of E α with the eigenvalue (1 + |k|) α . Consequently, for α > 0 we have N E α (λ) = #{k : (1 + |k|) α ≤ λ} #{k : |k| ≤ λ 1/α } = λ n/α , for the operator E α acting on Z n . Consequently, by Theorem 4.1, Part (i) and condition (5.24) we get that T ∈ S r (ℓ 2 (Z m ), ℓ 2 (Z n )) provided that 1 r < We note that integral operators on lattices can be also considered as pseudodifference operators, which is an analogue of pseudo-differential operators on the lattice Z n . Conditions for the membership of such operators in Schatten-von Neumann classes in terms of their symbols were given in [40]. 5.4. Conditions in terms of anharmonic oscillators. In Corollary 5.3 we considered the case of domains of R n of finite measure. We now discuss the case the whole space R n when the regularity of the kernel should be complemented by decay conditions at infinity.
For this, we consider a test with the anharmonic oscillator on L 2 (R n ), i.e., the operator E a = −∆ + |x| a on L 2 (R n ) for a > 0. For a = 2, the operator E 2 = −∆ + |x| 2 is the usual harmonic oscillator.
The study of the harmonic oscillator has been a very active field of research. For the spectral theory of non-commutative versions of the harmonic oscillator we refer to the interesting work of Parmeggiani et al [41], [42], [43], [44], [45], [46] and [47].
Since for operators on R n both the regularity and decay of the kernel at infinity are relevant it is natural to try to measure these properties of the kernel by the action of harmonic or anharmonic oscillators. The tests with anharmonic oscillators appear to be more natural when compared to the harmonic oscillator, since the orders of regularity and decay do not have to be the same. Moreover, it is natural to consider their fractional powers since the regularity or decay orders do not have to be integers.
Thus, as a consequence of the results of this paper we get the following conditions.
, then T belongs to the Schattenvon Neumann class S r (L 2 (R n ), L 2 (R m )) for all 0 < r < ∞ such that where p a = n( 1 a + 1 2 ) and p b = m( 1 b + 1 2 ). Moreover, The sequence of singular values (s k (T )) k satisfies the following estimate for the rate of decay: , and 1 2 < α pa + β p b , then T belongs to the trace class S 1 (L 2 (R n )) and its trace is given by Proof. The distribution of eigenvalues of E a and other second order differential operators has been investigated by E. C. Titchmarsh in [48]. In particular, Titchmarsh considered operators of the form −∆ + V (x) with V (x) → ∞ as |x| → ∞ and V (x) ultimately non-decreasing on every straight line radiating from the origin. If N(λ) denotes the number of eigenvalues less than λ, then he showed in [48,Section 17.8] that In particular if V (x) = |x| a we have .
We obtain N(λ) ∼ Cλ pa as λ → ∞, where p a = n( 1 a + 1 2 ). Now, since λ is an eigenvalue of E a if and only if λ α is an eigenvalue of (E a ) α , we obtain 27) where N P denotes the counting eigenvalue function for the operator P . Then we have The singular values of (E α a ) −1 are s k ((E α a ) −1 ) = λ −α k and (E α a ) −1 ∈ S q (L 2 (R n )) for all q > pa α . In a similar way we also have (E β b ) −1 ∈ S q ′ (L 2 (R m )) for all q ′ > p b β . As a consequence of Theorem 2.2 we obtain T ∈ S r (L 2 (R n ), L 2 (R m )) The rate of decay and Part (ii) now follow from Theorem 4.1.
Example 5.6. Let us give a simple example for Corollary 5.5: let T : L 2 (R n ) → L 2 (R n ) be an integral operator with kernel K(x, y). Assume that 1 ≤ n ≤ 3 and that ∆ x K, (1 + |x| b )K ∈ L 2 (R n × R n ). Then T is a trace class operator provided that b > 2n 4−n . This statement follows immediately from Part (ii) of Corollary 5.5 by taking α = 0, β = 1, implying that T is trace class provided that p b = n( 1 b + 1 2 ) < 2.

5.5.
Higher order anharmonic oscillators. One can also get a number of similar tests based on the estimation of N(λ) for different operators and the arguments in Corollary 5.5. Here we will consider different examples of anharmonic oscillators however restricting to integer orders of derivatives and weights. More specifically, let us consider the operator It is well known that such E has a discrete spectrum (see [35]) and it was also shown in [49,Theorem 3.2] that for large λ the eigenvalue counting function N(λ) is bounded by C a(x,ξ)<λ dxdξ, where a(x, ξ) is the Weyl symbol of the partial differential operator E. By the change of variables ξ = λ 1/2k ξ ′ and x = λ 1/2ℓ x ′ , we can estimate for large λ that We note that refined estimates for the remainder in the spectral asymptotics for N(λ) were also studied by Helffer and Robert in [50, Theorem 6 and Corollary 2.7] in the case k = ℓ, and in [51] for different k and ℓ in the case n = 1. Moreover, all the results remain unchanged if we add lower order terms to the operator E.
Consequently, from Theorem 4.1 and arguing similarly as in the proof of Corollary 5.5 we obtain: Let K ∈ L 2 (R n 2 × R n 1 ) and let T be the integral operator from L 2 (R n 1 ) to L 2 (R n 2 ) defined by Let α, β ≥ 0 and suppose that (E 2 ) β x (E 1 ) α y K ∈ L 2 (R n 2 × R n 1 ). Then T belongs to the Schatten-von Neumann class S r (L 2 (R n 1 ), L 2 (R n 2 )) for all 0 < r < ∞ such that Moreover, The sequence of singular values (s j (T )) j satisfies the following estimate for the rate of decay: ).
Example 5.8. Let us give a simple example for Corollary 5.7: let T : L 2 (R) → L 2 (R) be an integral operator with kernel K(x, y). Assume that k, l ∈ N and that K, d 2k dx 2k K, x 2l K ∈ L 2 (R × R). Then T ∈ S r (L 2 (R)) provided that 1 r < 1 2 + 2kl k+l . In particular, under the above assumptions T is always a trace class operator.
Remark 5.9. We would like now to consider the special case of a negative order and negative potential, more precisely the case of the hydrogen atom, i.e. an operator of the form H = −∆ − c|x| −1 on R 3 with c > 0. It is well known that the energy levels are of the form E n = − C n 2 where C is a positive constant. In this case we can take E −1 = H which belongs to the class S p with p > 1 2 . Therefore, one can obtain a similar result to Corollary 5.5 in terms of the operator E with an index p > 1 2 . 5.6. Subelliptic conditions on sub-Riemannian manifolds. In general, once the upper bound for the eigenvalue counting function of a certain operator is obtained, it can be used in Theorem 4.1. In particular, in some situations is may be convenient to use operators respecting certain geometric structures. Rather general results on the spectral asymptotics for self-adjoint subelliptic operators have been obtained by Fefferman and Phong [52,53] as well as for operators with double characteristics by Menikoff and Sjöstrand [54], see also an overview on spectral asymptotics for rather general hypoelliptic operators by Sjöstrand [55] and more recent extensions by Ponge [56] and Hassannezhad and Kokarev [57]. We can also refer to [58] for subelliptic analysis on nilpotent groups and to [59] for the potential theory for the sub-Laplacians.
Let We recall that the eigenvalue counting function of the sub-Laplacian on compact regular sub-Riemannian manifolds is estimated by N(λ) ≤ Cλ Q/2 , see e.g. [57]. Consequently, Theorem 4.1 immediately implies: Corollary 5.10. Let (M i , H i , g i ) (i = 1, 2) be compact regular sub-Riemannian manifolds and let ∆ i be the sub-Laplacians associated to H i . Let Q i denote the Hausdorff dimensions of M i with respect to the respective Carnot-Caratheodory distances.
Let K ∈ L 2 (M 2 × M 1 ) and let T be the integral operator from L 2 (M 1 ) to L 2 (M 2 ) defined by K(x, y)f (y)dy.
Let us briefly record two examples that are of particular importance: of compact contact manifolds and of compact Lie groups.
We recall that a contact manifold is a smooth manifold M of odd dimension 2n + 1 equipped with an 1-form θ such that θ∧(dθ) n is a volume form on M. The canonically induced bundle H x := {X ∈ T x M : θ(X) = 0} is regular and satisfies Hörmander's condition since 2-form dθ is non-degenerate on H. This will be the setting (C1) in the following statement. The setting (C2) concerns sub-Laplacians on compact Lie groups in which case the canonical sub-bundle is also regular due to the left-invariance.
Corollary 5.11. Let us consider the following situations: (C1) Let Ω i be a compact contact metric manifold of dimension 2n i + 1, (i = 1, 2). Let E i := (I + ∆ i ) α i for i = 1, 2, where ∆ i is the canonical positive sub-Laplacian on Ω i . Let p i := n i + 1, i = 1, 2. (C2) Let Ω i be a compact Lie group with left-invariant positive sub-Laplacian L i , and let Q i be the Hausdorff dimension of the induced control distance. Let E i := (I + L i ) α i and let p i := Q i 2 , i = 1, 2. Let K ∈ L 2 (Ω 2 × Ω 1 ) and let T be the integral operator from L 2 (Ω 1 ) to L 2 (Ω 2 ) defined by K(x, y)f (y)dµ 1 (y).