Boundedness of Pseudodifferential Operators on Banach Function Spaces

TL;DR

This paper establishes boundedness criteria for pseudodifferential operators on Banach function spaces, linking maximal operator boundedness to operator boundedness for specific symbol classes, with applications to variable Lebesgue spaces.

Contribution

It provides new boundedness results for pseudodifferential operators on Banach function spaces based on maximal operator bounds, extending to variable Lebesgue spaces.

Findings

Boundedness of pseudodifferential operators on Banach spaces under maximal operator conditions.

Extension of results to H"ormander and Miyachi symbol classes.

Application to variable Lebesgue spaces $L^{p( ext{·})}( ext{·})$.

Abstract

We show that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space $X(\mathbb{R}^n)$ and on its associate space $X'(\mathbb{R}^n)$, then a pseudodifferential operator $\operatorname{Op}(a)$ is bounded on $X(\mathbb{R}^n)$ whenever the symbol $a$ belongs to the H\"ormander class $S_{\rho,\delta}^{n(\rho-1)}$ with $0<\rho\le 1$, $0\le\delta<1$ or to the the Miyachi class $S_{\rho,\delta}^{n(\rho-1)}(\varkappa,n)$ with $0\le\delta\le\rho\le 1$, $0\le\delta<1$, and $\varkappa>0$. This result is applied to the case of variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R}^n)$.