Schatten classes on compact manifolds: Kernel conditions

In this paper we give criteria on integral kernels ensuring that integral operators on compact manifolds belong to Schatten classes. A specific test for nuclearity is established as well as the corresponding trace formulae. In the special case of compact Lie groups, kernel criteria in terms of (locally and globally) hypoelliptic operators are also given.


Introduction
Given a closed smooth manifold M (smooth manifold without boundary) endowed with a positive measure dx, in this paper we give sufficient conditions on Schwartz integral kernels in order to ensure that the corresponding integral operators belong to different Schatten classes. The problem of finding such criteria on different kinds of domains is classical and has been much studied, e.g. the paper [BS77] 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 this paper we present criteria for Schatten classes and, in particular, for the trace class operators on compact smooth manifolds without boundary. Compact Lie groups will also be considered as a special case since then additional results can be obtained, also allowing criteria in terms of hypoelliptic operators such as the sub-Laplacian. The sufficient conditions on integral kernels K(x, y) for Schatten classes will require regularity of a certain order in either x or y, or both.
We note that already some results of Birman and Solomyak [BS77] can be extended to compact manifolds but our approach allows one to be flexible about sets of variables in which one imposes the regularity of the kernel.
In order to obtain criteria for general Schatten classes we will use the well-known method of factorisation, particularly in the way applied by O'Brien in [O'B82]. For applications to trace formulas of Schrödinger operators see also [O'B83].
Schatten classes of pseudo-differential operators in the setting of the Weyl-Hörmander calculus have been considered in [Tof06], [Tof08], [BT10]. Schatten 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 [DR13b] and [DR13a]. In the subsequent part of the present paper we establish the characterisation of Schatten classes on closed manifolds in terms of symbols that we will introduce for this purpose.
In his classical book (cf. [Sug90], 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 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. However, in this paper we show that such type of results can be significantly improved by using a different approach. Associating a discrete Fourier Analysis to an elliptic operator on a compact manifold, we will establish the aforementioned connection in the setting of general closed manifolds, also weakening the known assumptions on the kernel for the operator to be trace class and for the trace formula (1.2) to hold. Thus, in this respect, a direct extension of the method employed by Sugiura leads to weaker results than our approach, for closed manifolds of dimension higher than 2, and we discuss this at the end of Section 4.
To formulate the notions more precisely, let H be a complex Hilbert space endowed with an inner product denoted by (·, ·), and let T : H → H be a linear compact operator. If we denote by T * : H → H the adjoint of T , then the linear operator (T * T ) 1 2 : H → H is positive and compact. Let (ψ k ) k be an orthonormal basis for H consisting of eigenvectors of |T | = (T * T ) 1 2 , and let s k (T ) be the eigenvalue corresponding to the eigenvector ψ k , k = 1, 2, . . . . The non-negative numbers s k (T ), k = 1, 2, . . . , are called the singular values of T : then the linear operator T : H → H is said to be in the trace class S 1 . It can be shown that S 1 (H) is a Banach space in which the norm · S 1 is given by multiplicities counted. 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. More generally, if 0 < p < ∞ and the sequence of singular values is p-summable, then T is said to belong to the Schatten class S p (H), and it is well known that each S p (H) is an ideal in L(H). If 1 ≤ p < ∞, a norm is associated to S p (H) by If 1 ≤ p < ∞ the class S p (H) becomes a Banach space endowed by the norm T Sp . If p = ∞ we define S ∞ (H) as the class of bounded linear operators on H, with T S∞ := T op , the operator norm. In the case 0 < p < 1 the quantity T Sp only defines a quasinorm, and S p (H) is also complete.
The Schatten classes are nested, with and satisfy the important multiplication property (cf. [Hor50], [Sim79], [GK69]) We will apply (1.4) for factorising our operators T in the form T = AB with A ∈ S p and B ∈ S q , and from this we deduce that T ∈ S r . A nice basic introduction to the study of the trace class is included in the book [Lax02] by Peter Lax. For the basic theory of Schatten classes we refer the reader to [GK69], [RS75], [Sim79], [Sch70].
In this paper we consider integral operators which is not restrictive in view of the Schwartz integral kernel theorem on closed manifolds. If H = L 2 (Ω, M, µ), it is well known that T is a Hilbert-Schmidt operator if and only if T can be represented by an integral kernel K = K(x, y) ∈ L 2 (Ω × Ω, µ ⊗ µ). In this paper we are interested in the case when Ω is a closed manifold (which we denote by M). In particular, we note that in view of (1.3) the condition K ∈ L 2 (M × M) implies that T ∈ S p for all p ≥ 2.
For p < 2, the situation is much more subtle, and the Schatten classes S p (L 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 [Car16] from 1916 gives the construction of a periodic continuous function κ(x) = ∞ k=−∞ c k e 2πikx for which the Fourier coefficients c k satisfy 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 φ k (x) = e 2πikx , T φ k = c k φ k . 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 (1.5) we have 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 class S p cannot be characterised by criteria of the type |K(x, y)| α dxdy < ∞, since the kernel K(x, y) = κ(x − y) of the operator T in (1.6) 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 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 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.
As for criteria for operators to belong to Schatten classes S p for 0 < p < 2, a simplified version of our results for kernels in Sobolev spaces can be stated as follows: for some µ > 0. Then the integral operator T on L 2 (M), defined by is in the Schatten classes S p (L 2 (M)) for p > 2n n+2µ . In particular, if µ > n 2 , then T is trace class.
This results improves, for example, Sugiura's result for the operator (1.1). Theorem 1.1 follows from the main result Theorem 3.6 giving criteria in terms of the mixed Sobolev spaces, Proposition 4.3, and Corollary 4.2. In particular, the use of mixed Sobolev spaces in Theorem 3.6 allows us to formulate criteria requiring different (smaller) regularities of K(x, y) in x and y, or only in one of these variables.
We note that the situation for Schatten classes S p for p > 2 is simpler and, in fact, similar to that of p = 2. For example, for left-invariant operators on compact Lie groups G, i.e. for convolution operators of the form T f = f * κ, it was shown in [DR13b] that the condition κ ∈ L p ′ (G), 1 ≤ p ′ ≤ 2, implies that T ∈ S p (L 2 (G)), where 1 p ′ + 1 p = 1. The converse of this is also true but for interchanged indices, i.e. the condition T ∈ S p (L 2 (G)) but now for 1 ≤ p ≤ 2 implies that κ ∈ L p ′ (G). We refer to [DR13b] for this as well as for the symbolic characterisation of Schatten classes in the setting of compact Lie groups.
In this work we allow singularities in the kernel so that the formula (1.2) 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 [Wei66] in the Euclidean setting, and applied by Brislawn in [Bri88], [Bri91] 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 [Del10a] and [Del10b]. 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 [Bri91] proved the following formula for a trace class operator T on L 2 (µ), for the extension to L p see [Del10a]: In Section 2 we describe the discrete Fourier analysis involved in our problem and establish several relations between eigenvalues and their multiplicities for elliptic positive pseudo-differential operators. Then in Section 3 we establish our criteria for Schatten classes on compact manifolds and, in particular, for the trace class in Section 4. For this, we briefly recall the definition of the averaging process involved in the formula (1.7). We also explain another method relating the convergence of the Fourier coefficients of the kernel with the traceability. In Section 5 the special case of compact Lie groups is considered where we show that the criteria can be also given using hypoelliptic operators.
The authors would like to thank Véronique Fischer and Jens Wirth for discussions and remarks.

Fourier analysis associated to an elliptic operator
In this section we start by recording some basic elements of Fourier analysis on compact manifolds which will be useful for our analysis.
Let M be a compact smooth manifold of dimension n without boundary, endowed with a fixed volume dx. We denote by Ψ ν (M) the Hörmander class of pseudodifferential operators of order ν ∈ R, i.e. operators which, in every coordinate chart, are operators in Hörmander classes on R n with symbols in S ν 1,0 , see e.g. [Shu01] or [RT10]. In this paper we will be using the class Ψ ν cl (M) of classical operators, i.e. operators with symbols having (in all local coordinates) an asymptotic expansion of the symbol in positively homogeneous components (see e.g. [Dui11]). Furthermore, we denote by Ψ ν + (M) the class of positive definite operators in Ψ ν cl (M), and by Ψ ν e (M) the class of elliptic operators in Ψ ν cl (M). Finally, Ψ ν +e (M) := Ψ ν + (M) ∩ Ψ ν e (M) will denote the class of classical positive elliptic pseudo-differential operators of order ν. We note that complex powers of such operators are well-defined, see e.g. Seeley [See67]. In fact, all pseudo-differential operators considered in this paper will be classical, so we may omit explicitly mentioning it every time, but we note that we could equally work with general operators in Ψ ν (M) since their powers have similar properties, see e.g. [Str72].
We now associate a discrete Fourier analysis to the operator E ∈ Ψ ν +e (M) inspired by those considered by Seeley ([See65], [See69]), see also [GW73]. However, we adapt it to our purposes and prove several auxiliary statements concerning the eigenvalues of E and their multiplicities, useful to us in the sequel.
The eigenvalues of E form a sequence {λ j }, with multiplicities taken into account. For each eigenvalue λ j , there is the corresponding finite dimensional eigenspace F j of functions on M, which are smooth due to the ellipticity of E. We set We also set d 0 := dim F 0 . Since the operator E is elliptic, it is Fredholm, hence also d 0 < ∞ (we can refer to [Ati68] for various properties of F 0 and d 0 ).
We fix an orthonormal basis of L 2 (M, dx) consisting of eigenfunctions of E: where {e k j } 1≤k≤d j is an orthonormal basis of F j . We denote by (·, ·) the standard inner product on L 2 (M) associated to its volume element. Let P j : L 2 (M) → F j be the corresponding orthogonal projections on F j . We observe that for f ∈ L 2 (M). The Fourier inversion formula takes the form for each f ∈ L 2 (M), and where the convergence is understood with respect to the L 2 (M)-norm.
For the distributional valuations we use the notation u(ϕ) or u, ϕ . If ϕ ∈ D(M) and u ∈ D ′ (M) we have We note that the same type of formula holds for operators from with an appropriate distributional understanding of convergence. The Fourier coefficients of u ∈ D ′ (M) associated to the basis (2.1) can be obtained from (2.3): In particular, if u ∈ L 2 (M) we obtain From the paragraph above we can deduce: in view of (2.3), completing the proof.
Comparing the Fourier inversion formula (2.2) in L 2 (M) with the formula (2.5) in D ′ (M), for a function f ∈ L 2 (M), we can identify its distributional Fourier coefficients f (j) with their action on the basis of F j given by . We will also denote sometimes by F the Fourier transform associating to f ∈ L 2 (M) its Fourier coefficients. Since forms a complete orthonormal system in L 2 (M), for all f ∈ L 2 (M) we have the Plancherel formula Since the criteria that we will obtain depend (a-priori) on the choice of the orthonormal basis {e k j }, the asymptotics of the corresponding eigenvalues play an essential role. We now establish several simple but useful relations between the eigenvalues λ j and their multiplicities d j .
Proposition 2.3. Let M be a closed manifold of dimension n, and let E ∈ Ψ ν +e (M), with ν > 0. Then there exists a constant C > 0 such that Proof. We observe that (1 + λ j ) 1/ν is an eigenvalue of the first-order elliptic positive operator (I + E) 1/ν of multiplicity d j . The Weyl formula for the eigenvalue counting function for the operator ( We now prove (2.8). Let us denote T : At the same time, by the functional calculus of pseudo-differential operators, we know that T ∈ Ψ −νq/2 (M), so that its kernel K(x, y) is smooth for x = y, and near the diagonal x = y, identifying points with their local coordinates, it satisfies the estimate for any α > n − νq/2, see e.g. [Dui11] or [RT10, Theorem 2.3.1], and the order is sharp with respect to the order of the operator. Thus, we get that K ∈ L 2 (M × M) if and only if we can choose α such that n > 2α > 2n − νq, which together with (2.9) implies (2.8).

Schatten classes on compact manifolds
Before stating our first result, we point out that a look at the proof of (1.2) (cf. [Sug90], Prop 3.5) shows that that statement can be already improved in the following way: Proposition 3.1. Let ∆ = ∂ 2 ∂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 4.
Our criteria for Schatten classes will also depend on a test of square integrability operating on the kernels through an elliptic operator, and the result of Proposition 3.1 will be improved in Theorem 3.6 (see specifically Corollary 4.2) by using a different approach to the problem. In the auxiliary next lemma we show that such condition is independent of the choice of an elliptic operator.
We first establish a simple observation for powers of positive elliptic operators to belong to Schatten classes S p on L 2 (M).
Proposition 3.3. Let M be a closed manifold of dimension n, and let E ∈ Ψ ν +e (M) be a positive elliptic pseudo-differential operator of order ν > 0. Let 0 < p < ∞. Then Proof. Let λ j denote the eigenvalues of E, each λ j having the multiplicity d j . Then the operator (I + E) −α is positive definite, its singular values are (1 + λ j ) −α with multiplicities d j . Therefore, which is finite if and only if αp > n ν by (2.8), implying the statement. If P is a pseudo-differential operator on M, for a function (or distribution) on M ×M, we will use the notation P y K(x, y) to indicate that the operator P is acting on the y-variable, the second factor of the product M ×M. For a positive elliptic operator P ∈ Ψ ν +e (M), by the elliptic regularity, the Sobolev space H µ (M) can be characterised as the space of all distributions f ∈ D ′ (M) such that (I + P ) µ ν f ∈ L 2 (M), and this characterisation is independent of the choice of operator P (see also Lemma 3.2).
We now define Sobolev spaces H µ 1 ,µ 2 x,y (M × M) of mixed regularity µ 1 , µ 2 ≥ 0. We observe that for K ∈ L 2 (M × M), we have or we can also write this as ). In particular, this means that K x defined by K x (y) = K(x, y) is well-defined for almost every x ∈ M as a function in L 2 y (M). Definition 3.4. Let K ∈ L 2 (M × M) and let µ 1 , µ 2 ≥ 0. We say that K ∈ By the elliptic regularity it follows that different choices of operators P ∈ Ψ ν +e (M), ν > 0, give equivalent norms on the space H µ 1 ,µ 2 x,y (M × M). Thus, for operators E j ∈ Ψ ν j +e (M) (j = 1, 2) with ν j > 0, we can formulate Definition 3.4 in an alternative (and perhaps more practical) way: , where the expression on the right hand side means that we are applying pseudodifferential operators on M separately in x and y. We note that these operators commute since they are acting on different sets of variables of K.
As we have noted above, the definition does not depend on a particular choice of operators E j ∈ Ψ ν j +e (M), with the norms of K induced by (3.3) being all equivalent to each other and to that in Definition 3.4. In Proposition 4.3 we establish some properties of the spaces H µ 1 ,µ 2 x,y , namely, we will show the inclusions between the mixed and the standard Sobolev spaces on the compact (closed) manifold M × M as , for any µ 1 , µ 2 ≥ 0.
We will now give our main criteria for Schatten classes.
Remark 3.7. The value for r comes from the relation for some 0 < p 1 , p 2 < ∞, where the condition r > 2n n+2(µ 1 +µ 2 ) comes from µ j > n p j ν j by a suitable application of (3.1). Also, since then r = 2p 1 p 2 p 1 p 2 +2(p 1 +p 2 ) , the range for r is the interval (0, 2) since, in general, 0 < p j < ∞. Therefore, Theorem 3.6 provides a sufficient condition for Schatten classes S r for 0 < r < 2. For µ 1 , µ 2 = 0 the conclusion is trivial and can be sharpened to include r = 2.
Remark 3.8. We note that for µ 1 = 0, Theorem 3.6 says that for K ∈ L 2 (M, H µ (M)), we have that the corresponding operator T satisfies T ∈ S r for r > 2n n+2µ . In this case no regularity in the x-variable is imposed on the kernel.
We also note that the 'dual' result with µ 2 = 0 imposing no regularity of K with resect to y also follows directly from it by considering the adjoint operator T * and using the equality T * Sr = T Sr .
Proof of Theorem 3.6. Let, for example, E = (I + ∆ M ) 1 2 , where ∆ M is a positive definite elliptic differential operator of order 2, and E = E. The existence of such ∆ M follows, for example, from the Whitney embedding theorem.
First we note that by (3.3) the condition K ∈ H µ 1 ,µ 2 x,y (M × M) can be written as E µ 1 x E µ 2 y K ∈ L 2 (M × M), with ν 1 = ν 2 = 1. Since E µ 1 x E µ 2 y K ∈ L 2 (M × M), we have E µ 1 x E µ 2 y K(x, ·) ∈ L 2 y (M) for almost every x, and this fact will justify the use of scalar products in the next argument.
(ii) If µ 1 = 0 and µ 2 > 0, just by removing the operator E µ 1 x in the argument above we get the desired result. (iii) If µ 1 > 0 and µ 2 = 0. This is a consequence of case (ii) proceeding by duality, considering the adjoint of the operator T and applying the fact that T Sr = T * Sr .

Trace class operators and their traces
We shall 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 that case, if f ∈ L p (µ), then E(f |M n ) exists. We say that a sequence {f j } j of functions on Ω is a martingale if each f j is M j -measurable and 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, on a second countable topological space endowed with 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 with Ω = M, the closed manifold. Now, for each (x, y) ∈ M × M 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 [Doo94] for more details on the martingale maximal function and its properties.
We denote by A (2) j the averaging operators on Ω × Ω:

K(s, t)dµ(t)dµ(s).
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 operators, then K(x, y) is, in particular, square integrable, and hence by the Hölder inequality it is integrable on the compact manifold M × M.
In the sequel in this section, we can always assume that K ∈ L 2 (M × M) 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.
As a corollary of Theorem 3.6, for the trace class operators we have: Proof. We observe that to get r = 1 from Theorem 3.6, we require the following inequality to hold: .
But this is equivalent to µ 1 + µ 2 > n 2 . The trace formula is a consequence of (1.7). Corollary 4.1 improves, in particular, Proposition 3.1: Then the integral operator T on L 2 (M), defined by is trace class on L 2 (M) and its trace is given by (4.4).  Proof. Let ∆ be a second order positive elliptic differential operator on M (such an operator exists e.g. by the Whitney embedding theorem), and let e k j denote the orthonormal basis and λ j the corresponding eigenvalues, leading to the discrete Fourier analysis associated to ∆ as described in Section 2. Then the products e k j (x)e m l (y) give rise to an orthonormal basis in L 2 (M × M). Consequently, using (3.3) with E 1 = E 2 = ∆, we see that K ∈ H µ 1 ,µ 2 x,y (M × M) is equivalent to the condition where the Fourier coefficients K(j, k, l, m) are determined by the Fourier series K(j, k, l, m)e k j (x)e m l (y).
We also obtain some corollaries in terms of the derivatives of the kernel. We denote by C α x C β y (M × M) the space of functions of class C β with respect to y and C α with respect to x.
Corollary 4.4. Let M be a closed manifold of dimension n. Let K ∈ C ℓ 1 x C ℓ 2 y (M ×M) some even integers ℓ 1 , ℓ 2 ∈ 2N 0 such that ℓ 1 + ℓ 2 > n 2 . Then the integral operator T on L 2 (M), defined by is in S 1 (L 2 (M)), and its trace is given by Proof. Let ∆ M be an elliptic positive definite second order differential operator on . Now, by observing that ℓ 1 + ℓ 2 > n 2 the result follows from Corollary 4.1 . Remark 4.5. The index n 2 in Corollary 4.4 is sharp. Indeed, for the torus T n with n even, there exist a function χ of class C n 2 such that the series of Fourier coefficients diverges (cf. [SW71], Ch. VII; [Wai65]). 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. Hence, T / ∈ S 1 (L 2 (T n )) but K ∈ C n 2 (M × M) (we can think of e.g. ℓ 1 = 0 and ℓ 2 = n 2 in Corollary 4.4). On the other hand, concerning necessary conditions on the kernel, writing the convolution operator T in the pseudo-differential form with σ(ξ) = χ(ξ), it can be shown that ξ∈Z n |σ(ξ)| < ∞ if and only if the corresponding pseudo-differential operator T σ is trace class on L 2 (T n ) (cf. [DR13b]). Hence, when dealing with a multiplier we can deduce that if T σ is trace class then its kernel is continuous. This can be obtained from the formula for the convolution kernel K(x, y) = ξ∈Z n e i(x−y)ξ σ(ξ) and the summability of σ. Therefore, the continuity of kernels is a necessary condition for traceability of convolution operators on T n . However, as we note from the example (1.6) on M = T 1 the convolution kernel K(x, y) = κ(x − y) is continuous but the corresponding operator is not trace class.
We now make some remarks about the relation between the trace class property and the Fourier coefficients of the kernel, in the sense of Section 2.
The main idea in the proof by Sugiura of (1.2) (and then also of Proposition 3.1) consists in exploiting the underlying relation between the convergence of the series of Fourier coefficients of the kernel K ∈ C 2 (T 1 × T 1 ) and the traceability. This link is basically reduced to the application of the following classical result, see e.g. [Sug90]: Lemma 4.6. Let H be a separable Hilbert space. If a bounded linear operator T on H satisfies m,j∈N for a fixed orthonormal basis (φ j ) j , then T is trace class.
By choosing an orthonormal basis {φ j } consisting of eigenfunctions of the Laplacian on T 1 × T 1 one can prove that the Fourier coefficients of K agree with the values (T φ j , φ m ). On the other hand, it can be shown that the series of Fourier coefficients converges absolutely and hence the traceability follows. The proof can be extended to smooth closed manifolds by using elliptic positive pseudo-differential operators instead of Laplacians on M × M, and associating a discrete Fourier analysis to the cross product M × M. However, this method leads to a weaker result, by furnishing the condition K = K(x, y) ∈ C ν (M × M) with ν > n. We have improved that kind of result by obtaining a sharp condition on the regularity of K with respect to y as given in Corollary 4.4 (with ℓ 1 = 0 and ℓ 2 > n/2).
However, for the sake of completeness we establish below a result concerning the convergence of series of Fourier coefficients. The related problem concerning the convergence of Fourier series on compact manifolds has been studied by Taylor [Tay68]. Taylor's paper also included a special version for compact Lie groups. Similar results for compact connected Lie groups from a different approach were obtained by Sugiura in [Sug71]. In order to study such kind of convergence for a kernel K on M × M we will first apply the convergence criterion (2.8) to the manifold M × M. In the following lemma we will associate to an elliptic positive pseudo-differential operator For the Fourier coefficients of K, by the Cauchy-Schwarz inequality, (4.10) and the inequality (2.8) applied to E on M × M we obtain where for the application of (2.8), we note that the convergence of the series ℓ d ℓ (1 + λ ℓ ) −2 on M ×M with q = 2 is equivalent to the condition ν > n. The independence of the choice of an elliptic positive pseudo-differential operator of order ν is an immediate consequence of Lemma 3.2 for the manifold M × M.

Schatten classes on compact Lie groups
In this section we consider the conditions for Schatten classes for operators on compact Lie groups. We show that the conditions on the kernel can be also formulated in terms of hypoelliptic operators. This is done by combining the factorisation method used in the previous sections with recent results [DR13b] by the authors on characterisation of invariant operators in Schatten classes on compact Lie groups. We start by describing the basic concepts we will require for this setting.
Let G be a compact Lie group of dimension n with the normalised Haar measure dx. Let G denote the set of equivalence classes of continuous irreducible unitary representations of G. Since G is compact, the set G is discrete. For [ξ] ∈ G, by choosing a basis in the representation space of ξ, we can view ξ as a matrix-valued function ξ : G → C d ξ ×d ξ , where d ξ is the dimension of the representation space of ξ. By the Peter-Weyl theorem the collection is the orthonormal basis of L 2 (G). If f ∈ L 1 (G) we define its global Fourier transform at ξ by If ξ is a matrix representation, we have f (ξ) ∈ C d ξ ×d ξ . The Fourier inversion formula is a consequence of the Peter-Weyl theorem, and we have For each [ξ] ∈ G, the matrix elements of ξ are the eigenfunctions for the Laplacian L G (or the Casimir element of the universal enveloping algebra), with the same eigenvalues which we denote by −λ 2 [ξ] , so that we have The weight for measuring the decay or growth of Fourier coefficients in this setting is ξ := (1 + λ 2 [ξ] ) 1 2 , the eigenvalues of the (positive) elliptic first-order pseudodifferential operator (I − L G ) 1 2 . The Parseval identity takes the form which gives the norm on ℓ 2 ( G).

For a linear continuous operator
Then one has ( [RT10], [RT13]) the global quantization in the sense of distributions, and the sum is independent of the choice of a representation ξ from each equivalence class [ξ] ∈ G. If A is a linear continuous operator from C ∞ (G) to C ∞ (G), the series (5.4) is absolutely convergent and can be interpreted in the pointwise sense. We will also write A = Op(σ A ) for the operator A given by the formula (5.4). We refer to [RT10,RT13] for the consistent development of this quantization and the corresponding symbolic calculus.
In the subsequent Part II of this paper, we will relate the Fourier and symbolic analysis on general closed manifolds to those on compact Lie groups. So, now we will only concentrate on showing that the conditions for Schatten classes can be also formulated by examining the regularity of the kernel under the action of non-elliptic but hypoelliptic operators.
Instead of Proposition 3.3 for elliptic operators as the starting point, here we will use its analogue for hypoelliptic operators established in [DR13b], for example its analogue for sub-Laplacians on compact Lie groups G. Let us write L G = X 2 1 + · · · + X 2 n−1 + X 2 n , for a basis X 1 , . . . , X n of left-invariant vector fields on the Lie algebra g of G, assuming that the span of the first commutators of X 1 , . . . , X n−1 contains X n . Then it was shown in [DR13b] that (5.6) 0 < r < ∞ and αr > 2n =⇒ (I − L sub ) −α/2 ∈ S r (L 2 (G)).
Here we can note that the powers of the hypoelliptic positive pseudo-differential operator I − L sub are well-defined. There is a general theory, see e.g. [KgT73], [HKg71], or more recent results and references in [BN07]. However, if we observe that the matrix symbol of the sub-Laplacian (as well as the symbol of the operator H γ is the sequel) are diagonal, a complex power of such operator may be defined by the quantization formula (5.4) using the matrix symbol being the corresponding complex power of the diagonal symbol of the operator. For left-invariant operators with diagonal matrix symbols (such as L sub or H γ ) all such approaches yield the same operators (see e.g. [RW13] for more details). Consequently, arguing in the same way as in the proof of Theorem 3.6 we obtain: Corollary 5.1. Let G be a compact Lie group of dimension n, and let L sub be a sub-Laplacian as in (5.5). Let K ∈ L 2 (M ×M) be such that (I −L sub ) µ 1 /2 x (I −L sub ) µ 2 /2 y K ∈ L 2 (G × G) for some µ 1 , µ 2 ≥ 0. Then the integral operator T on L 2 (G), defined by is in the Schatten classes S r (L 2 (G)) for r > 2n n+(µ 1 +µ 2 ) . Proof. We argue similar to the proof of Theorem 3.6. In particular, we know from (5.6) that (I − L sub ) −µ j /2 ∈ S p j for µ j p j > 2n (j = 1, 2). From the relation 1 r = 1 2 + 1 p 1 + 1 p 2 and p j > 2n µ j , we get that under the condition r > 2n n+(µ 1 +µ 2 ) the operator T belongs to the Schatten class S r on L 2 (G).
As it was noted in [DR13b], the implication (5.6) can be improved for particular groups using their particular structure. For example, for the compact Lie group SU(2) we have, for three left-invariant vector fields X, Y, Z that [X, Y ] = Z, and so with L sub = X 2 + Y 2 we have (5.7) 0 < r < ∞ and αr > 4 =⇒ (I − L sub ) −α/2 ∈ S r (L 2 (SU(2))).
We now show that instead of the sub-Laplacian other globally hypoelliptic operators can be used, also those that are not necessarily covered by Hörmander's sum of the squares theorem. Instead of SU(2), for a change, we will formulate this for the group SO(3) noting that, however, the same conclusion holds also on SU(2) ≃ S 3 . To formulate and motivate the result, we first briefly introduce some more notation concerning the group G = SO (3)  The dimension of each t ℓ is d t ℓ = 2ℓ + 1.
As in the case of SU(2), let us fix three left-invariant vector fields X, Y, Z on SO(3) associated to the derivatives with respect to the Euler angles, so that we also have [X, Y ] = Z, see [RT10] or [RT13] for the detailed expressions.
The same conclusion holds on SU(2) ≃ S 3 . Again, if µ 1 = µ 2 = 0, the results have a trivial strengthening to include the case r = 2.