Abstract
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 Rn of finite measure, and conditions for operators on Rn 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.
Highlights
Let (Ωj, Mj, μj) (j = 1, 2) be measure spaces respectively endowed with a σ-finite measure μj on a σ-algebra Mj of subsets of Ωj
In this paper we give sharp sufficient conditions on integral kernels K = K(x, y) in order to ensure that the corresponding integral operators
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
Summary
|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 90 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 Schattenvon Neumann classes for p < 2 and, in particular, motivates the results in this paper. 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 This process is most effective for the computations in the case of a σ-algebra of Borel sets for a second countable topological space. Let K ∈ L2(μ2 ⊗ μ1) and let T be the integral operator from L2(μ1) to L2(μ2) defined by (i) If (E2)x(E1)yK ∈ L2(μ2⊗μ1), T belongs to the Schatten-von Neumann class Sr(L2(μ1), L2(μ2)). 1 11 < +, rq p and respectively (2.9) or (2.10) holds
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