Schatten classes, nuclearity and nonharmonic analysis on compact manifolds with boundary

TL;DR

This paper develops a symbolic calculus for pseudo-differential operators on compact manifolds with boundary, including singular boundaries, and provides criteria for Schatten class membership and nuclearity, extending classical analysis tools.

Contribution

It introduces a global symbolic calculus for boundary operators with arbitrary boundary singularities and establishes new criteria for Schatten class and nuclearity on manifolds with boundary.

Findings

Criteria for Schatten class membership on $L^2(M)$

Conditions for $r$-nuclearity on $L^p(M)$

Extension of Grothendieck-Lidskii formula in this setting

Abstract

Given a compact manifold $M$ with boundary $\partial M$, in this paper we introduce a global symbolic calculus of pseudo-differential operators associated to $(M,\partial M)$. The symbols of operators with boundary conditions on $\partial M$ are defined in terms of the biorthogonal expansions in eigenfunctions of a fixed operator $L$ with the same boundary conditions on $\partial M$. The boundary $\partial M$ is allowed to have (arbitrary) singularities. As an application, several criteria for the membership in Schatten classes on $L^2(M)$ and $r$-nuclearity on $L^p(M)$ are obtained. We also describe a new addition to the Grothendieck-Lidskii formula in this setting. Examples and applications are given to operators on $M=[0,1]^n$ with non-periodic boundary conditions, and of operators with non-local boundary conditions.