Kernel and symbol criteria for Schatten classes and $r$-nuclearity on compact manifolds

In this note we present criteria on both symbols and integral kernels ensuring that the corresponding 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. A notion of an invariant operator and its full symbol associated to an elliptic operator are introduced. Some applications to the study of $r$-nuclearity on $L^{p}$ spaces are also obtained.


Introduction
Let M be a closed smooth manifold (smooth manifold without boundary) endowed with a positive measure dx. We denote by Ψ ν (M) the usual Hörmander class of pseudo-differential operators of order ν ∈ R. In this paper we will be using the class Ψ ν cl (M) of classical operators (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 associate a discrete Fourier analysis to the operator E ∈ Ψ ν +e (M) inspired by that considered by Seeley ([See65], [See69]), see also [GW73]. The eigenvalues of E form a sequence {λ j } 0≤j<∞ , with λ 0 := 0 and with multiplicities d j . The corresponding orthonormal basis of L 2 (M, dx) consisting of eigenfunctions of E will be denoted by {e k j } 1≤k≤d j j≥0 . Relative to this basis, Fourier coefficients, Plancherel identity and Fourier inversion formula can be obtained.
By introducing suitable Sobolev spaces on M × M adapted to E we give first sufficient Sobolev type conditions on Schwartz integral kernels in order to ensure that the corresponding integral operators belong to different Schatten classes. Second, we introduce notions of invariant operators, Fourier multipliers and full matrix-symbols relative to E. We apply those notions to characterise Schatten classes and to find sufficient conditions for nuclearity and r-nuclearity in the sense of Grothendieck [Gro55] for invariant operators in terms of the full matrix-symbol.
Date: August 27, 2014. The first author was supported by Marie Curie IIF 301599. The second author was supported by EPSRC grant EP/K039407/1.

Kernel conditions for Schatten classes on compact manifolds
We first define Sobolev spaces H µ 1 ,µ 2 x,y (M × M) of mixed regularity µ 1 , µ 2 ≥ 0. Definition 2.1. Let K ∈ L 2 (M ×M) and let µ 1 , µ 2 ≥ 0. For operators E j ∈ Ψ ν j +e (M) (j = 1, 2) with ν j > 0, we define 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.
By the elliptic regularity the spaces H µ 1 ,µ 2 x,y (M × M) do not depend on a particular choice of E 1 , E 2 as above. We will now give our main kernel criterion obtained in [DR14c] for Schatten classes.
Theorem 2.2. Let M be a closed manifold of dimension n and let µ 1 , µ 2 ≥ 0. Let K ∈ L 2 (M × M) be such that K ∈ H µ 1 ,µ 2 x,y (M × M). Then the corresponding integral operator T K on L 2 (M) given by T K f (x) = M K(x, y)f (y)dy is in the Schatten classes S r (L 2 (M)) for r > 2n n+2(µ 1 +µ 2 ) . We formulate below the results on the trace class. Due to possible singularities of the kernel K(x, y) along the diagonal, we will require an averaging process on K as described in [DR14c] (see also [Bri91], [Del10]). We denote by K(x, y) the pointwise values of such process defined a.e. As a corollary of Theorem 2.2, for the trace class we have: Corollary 2.3. Let M be a closed manifold of dimension n and let K ∈ L 2 (M × M), µ 1 , µ 2 ≥ 0, be such that µ 1 +µ 2 > n 2 and K ∈ H µ 1 ,µ 2 x,y (M ×M). Then the corresponding integral operator T K on L 2 (M) is trace class and its trace is given by (2.2) We also obtain several 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.
, and its trace is given by (2. 3) The corollary above is sharp (cf. Remark 4.5 of [DR14c]) as a consequence of classical results for the convergence of Fourier series on the torus (cf. [SW71], Ch. VII; [Wai65]).
We now formulate some consequences on compact Lie groups combining results from [DR13]. For example, on the compact Lie group SU(2) let X, Y, Z be three left-invariant vector fields X, Y, Z such that [X, Y ] = Z (for example, these would be derivatives with respect to Euler angles at a point extended to the whole of SU(2) by the left-invariance). Let L sub = X 2 + Y 2 be the sub-Laplacian. Then we have 0 < r < ∞ and αr > 4 =⇒ (I − L sub ) −α/2 ∈ S r (L 2 (SU(2))).
(2.4) The same is true for S 3 ≃ SU(2) considered as the compact Lie group with the quaternionic product. Using this instead of elliptic operators we can show: 2+µ 1 +µ 2 . The same result holds on the compact Lie groups SU(2) and SO(3). We now argue 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. We will formulate this for the group SO(3) noting that, however, the same conclusion holds also on SU(2) ≃ S 3 . We 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. We consider the following family of 'Schrödinger' differential operators for a parameter 0 < γ < ∞. For γ = 1 it was shown in [RTW14] that H 1 + cI is globally hypoelliptic if and only if 0 ∈ {c+ℓ(ℓ+1)−m(m+1) : ℓ ∈ N, m ∈ Z, |m| ≤ ℓ}. It has been also shown in [DR13, Section 4] that, if γ > 1, then I + H γ is globally hypoelliptic, and (I + H γ ) −α/2 ∈ S p if and only if αp > 4. As a consequence of this and following the argument in [DR14c] for the proof Theorem 2.2 with I + H γ instead of E = ∆ M for the manifold M = SO(3), as well as Corollary 2.5, we obtain: Corollary 2.6. Let K ∈ L 2 (SO(3)×SO(3)) be such that (I +H γ ) (3)) for some µ 1 , µ 2 ≥ 0. Then the integral operator T K on L 2 (SO(3)) is in S r for r > 4 2+µ 1 +µ 2 and γ > 1.

Symbols, Fourier multipliers and nuclearity
Let us now consider the concepts of invariant operator and corresponding full symbols introduced in [DR14a]. The eigenvalues of E ∈ Ψ ν +e (M) (counted without multiplicities) form a sequence {λ j } which we order so that 0 = λ 0 < λ 1 < λ 2 < · · · . (3.1) For each eigenvalue λ j , there is the corresponding finite dimensional eigenspace H j of functions on M, which are smooth due to the ellipticity of E. We set d j := dim H j , and H 0 := ker E.
We also set d 0 := dim H 0 . Since the operator E is elliptic, it is Fredholm, hence also d 0 < ∞ (we can refer to [Ati68], [Hör85a] for various properties of H 0 and d 0 ). We fix an orthonormal basis of L 2 (M) consisting of eigenfunctions of E: where {e k j } 1≤k≤d j is an orthonormal basis of H j . Let P j : L 2 (M) → H j be the corresponding projection.
The Fourier coefficients of f ∈ L 2 (M) with respect to the orthonormal basis {e k j } will be denoted by We will call the collection of f (j, k) the Fourier coefficients of f relative to E, or simply the Fourier coefficients of f . If f ∈ L 2 (M), we also write thus thinking of the Fourier transform always as a column vector. The following theorem proved in [DR14a] is the base to introduce the concepts of invariant operators and full symbols relative to E.
Theorem 3.1. Let M be a closed manifold and let T : C ∞ (M) → L 2 (M) be a linear operator. Then the following conditions are equivalent: for all f ∈ C ∞ (M). The matrices σ(ℓ) in (iii) and (iv) coincide. If T extends to a linear continuous operator T : D ′ (M) → D ′ (M) then the above properties are also equivalent to the following ones: (v) For each j ∈ N 0 , we have T P j = P j T on C ∞ (M).
If any of the equivalent conditions (i)-(iv) of Theorem 3.1 are satisfied, we say that the operator T : C ∞ (M) → L 2 (M) is invariant (or is a Fourier multiplier) relative to E. We can also say that T is E-invariant or is an E-multiplier. When there is no risk of confusion we will just refer to such kind of operators as invariant operators or as multipliers. If T extends to a linear continuous operator T : D ′ (M) → D ′ (M) then we will say that T is strongly invariant relative to E.
The proposition below shows how invariant operators can be expressed in terms of their symbols.
We can now formulate our characterisation of the membership of invariant operators in Schatten classes: If an invariant operator T : L 2 (M) → L 2 (M) is in the trace class S 1 (L 2 (M)), then Tr(σ T (ℓ)).
We now turn to some applications to the nuclearity on L p (M) spaces. Let F 1 and F 2 be two Banach spaces and 0 < r ≤ 1, a linear operator T from F 1 into F 2 is called r-nuclear if there exist sequences (x ′ n ) in F ′ 1 and (y n ) in F 2 so that This notion, developed by Grothendieck [Gro55], extends the notion of Schatten classes to the setting of Banach spaces.
In order to study nuclearity on L p (M) for a given compact manifold M of dimension n, we introduce a function Λ(j, k; n, p) which controls the L p -norms of the family of eigenfunctions {e k j } of the operator E, i.e. we will suppose that Λ(j, k; n, p) is such that we have the estimates e k j L p (M ) ≤ Λ(j, k; n, p). (3.7) There are many things that can be said about the behaviour of Λ(j, k; n, p) in different settings, see e.g. results and discussions in [Don06,TZ02,DR14a]. We will use the following functionp for 1 ≤ p ≤ ∞: (3.8) For 1 ≤ p 1 , p 2 ≤ ∞ we denote their dual indices by q 1 := p ′ 1 , q 2 := p ′ 2 . The criterion for r-nuclearity now is: Theorem 3.4. Let 1 ≤ p 1 , p 2 < ∞ and 0 < r ≤ 1. Let T : L p 1 (M) → L p 2 (M) be a strongly invariant linear continuous operator. Assume that its matrix-valued symbol σ(ℓ) satisfies ∞ ℓ=0 d ℓ m,k=1 |σ(ℓ) mk | r Λ(ℓ, m; n, ∞)p 2 r Λ(ℓ, k; n, ∞)q 1 r < ∞.
Then the operator T : L p 1 (M) → L p 2 (M) is r-nuclear.
We now give an example of the application of such results in the case of the sphere S 3 ≃ SU(2). We consider the Laplacian (the Casimir element) E = −L S 3 .
The first author was supported by Marie Curie IIF 301599. The second author was supported by EPSRC grant EP/K039407/1. The authors would like to thank Véronique Fischer for discussions.