Fourier multipliers, symbols and nuclearity on compact manifolds

TL;DR

This paper develops a framework for Fourier multipliers on compact manifolds, characterizing invariant operators, analyzing Schatten classes, and establishing conditions for r-nuclearity between Lp-spaces.

Contribution

It introduces a symbol notion for invariant operators on compact manifolds and applies it to Schatten class characterization and nuclearity conditions.

Findings

Characterization of Fourier multipliers on compact manifolds.

Formula for the trace of trace class operators.

Conditions for operators to be r-nuclear between Lp-spaces.

Abstract

The notion of invariant operators, or Fourier multipliers, is discussed for densely defined operators on Hilbert spaces, with respect to a fixed partition of the space into a direct sum of finite dimensional subspaces. As a consequence, given a compact manifold endowed with a positive measure, we introduce a notion of the operator's full symbol adapted to the Fourier analysis relative to a fixed elliptic operator. We give a description of Fourier multipliers, or of operators invariant relative to the elliptic operator. We apply these concepts to study Schatten classes of operators and to obtain a formula for the trace of trace class operators. We also apply it to provide conditions for operators between Lp-spaces to be r-nuclear in the sense of Grothendieck.