Pseudo-differential operators in H\"older spaces revisited. Weyl-H\"ormander calculus and Ruzhansky-Turunen classes

TL;DR

This paper establishes continuity results for pseudo-differential operators in H"older spaces within the Weyl-H"ormander calculus, extending classical boundedness theorems and analyzing Ruzhansky-Turunen classes.

Contribution

It provides new continuity results for operators in the Weyl-H"ormander calculus on H"older spaces and explores their action within Ruzhansky-Turunen classes.

Findings

Recovered classical H"older boundedness theorems.

Extended the analysis to hypoelliptic Laplacians.

Investigated the action of Ruzhansky-Turunen classes on H"older spaces.

Abstract

In this work we obtain continuity results on H\"older spaces for operators belonging to a Weyl-H\"ormander calculus for metrics such that the class of the associated operators contains, in particular, certain hypoelliptic laplacians. With our results we recover some historical H\"older boundedness theorems (see R. Beals \cite{Be,Be2}). The action of (periodic) Ruzhansky-Turunen classes of pseudo-differential operators on H\"older spaces also will be investigated.