$L^p$ and Sobolev boundedness of pseudodifferential operators with non-regular symbol: a regularisation approach

TL;DR

This paper studies the boundedness of pseudodifferential operators with non-regular symbols in $L^p$ and Sobolev spaces, using regularisation techniques to establish estimates that connect symbol irregularity with operator behavior.

Contribution

It introduces a regularisation approach employing mollifiers to analyze $L^p$ and Sobolev boundedness of pseudodifferential operators with non-regular symbols.

Findings

Established $L^p$ bounds for regularised operators

Derived Sobolev estimates linked to symbol regularity

Connected regularisation parameter with operator boundedness

Abstract

In this paper we investigate $L^p$ and Sobolev boundedness of a certain class of pseudodifferential operators with non-regular symbols. We employ regularisation methods, namely convolution with a net of mollifiers $(\rho_\eps)_\eps$, and we study the corresponding net of pseudodifferential operators providing $L^p$ and Sobolev estimates which relate the parameter $\eps$ with the non-regularity of the symbol.