Schatten-von Neumann classes of integral operators

TL;DR

This paper provides sharp kernel conditions that determine when integral operators belong to Schatten-von Neumann classes, with applications across various geometric and spectral settings.

Contribution

It introduces new criteria based on spectral properties of kernels, extending Schatten class characterizations to diverse geometric contexts.

Findings

Established sharp kernel conditions for Schatten class membership.

Derived spectral criteria for differential operators in various settings.

Provided examples on manifolds, Lie groups, and sub-Riemannian structures.

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 ${\mathbb R}^n$ 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.