Pseudodifferential operators associated with a semigroup of operators

TL;DR

This paper develops a framework for pseudodifferential operators linked to semigroups on various metric measure spaces, establishing boundedness results and analyzing symbol classes, including on Riemannian manifolds and fractals.

Contribution

It introduces a general approach to pseudodifferential operators associated with semigroups on diverse metric spaces, extending classical theory to new settings.

Findings

Boundedness of order 0 pseudodifferential operators on L^p spaces.

Analysis of symbols in the class S^0_{1,δ} for δ in [0,1).

Results on the class S^0_{1,1} for sub-Laplacians on Riemannian manifolds.

Abstract

Related to a semigroup of operators on a metric measure space, we define and study pseudodifferential operators (including the setting of Riemannian manifold, fractals, graphs ...). Boundedness on $L^p$ for pseudodifferential operators of order 0 are proved. Mainly, we focus on symbols belonging to the class $S^0_{1,\delta}$ for $\delta\in[0,1)$. For the limit class $S^0_{1,1}$, we describe some results by restricting our attention to the case of a sub-Laplacian operator on a Riemannian manifold.