Pointwise estimates of pseudo-differential operators

TL;DR

This paper introduces a new technique for pointwise estimates of pseudo-differential operators acting on functions with compact spectra, utilizing a factorisation inequality involving maximal functions and symbol factors, with applications to $L_p$ bounds.

Contribution

It presents a novel pointwise estimation method for pseudo-differential operators using factorisation inequalities, enabling new bounds and insights, especially for type $1,1$-operators.

Findings

Establishes pointwise estimates using Peetre--Fefferman--Stein maximal functions.

Provides bounds for pseudo-differential operators in $L_p$ spaces.

Shows type $1,1$-operators are bounded on $L_p$-functions with compact spectra.

Abstract

As a new technique it is shown how general pseudo-differential operators can be estimated at arbitrary points in Euclidean space when acting on functions $u$ with compact spectra. The estimate is a factorisation inequality, in which one factor is the Peetre--Fefferman--Stein maximal function of $u$, whilst the other is a symbol factor carrying the whole information on the symbol. The symbol factor is estimated in terms of the spectral radius of $u$, so that the framework is well suited for Littlewood--Paley analysis. It is also shown how it gives easy access to results on polynomial bounds and estimates in $L_p$, including a new result for type $1,1$-operators that they are always bounded on $L_p$-functions with compact spectra.