$L_p$-Theory of Type $1,1$-Operators

TL;DR

This paper extends the theory of type 1,1 pseudo-differential operators, proving their continuity across various function spaces using paradifferential decomposition and maximal function estimates, with special focus on self-adjoint operators.

Contribution

It provides new continuity results for type 1,1 pseudo-differential operators in multiple function spaces, including Besov and Triebel--Lizorkin spaces, and extends these results to arbitrary smoothness for self-adjoint subclasses.

Findings

Proved continuity in $L_p$-Sobolev, H"older--Zygmund, Besov, and Triebel--Lizorkin spaces.

Extended results to arbitrary smoothness for self-adjoint operators.

Utilized paradifferential decomposition and maximal function estimates as main tools.

Abstract

This is a continuation of recent work on the general definition of pseudo-differential operators of type $1,1$, in H\"ormander's sense. Continuity in $L_p$-Sobolev spaces and H\"older--Zygmund spaces, and more generally in Besov and Lizorkin--Triebel spaces, is proved for positive smoothness, with extension to arbitrary smoothness for operators in the self-adjoint subclass. As a main tool the paradifferential decomposition is used for type $1,1$-operators in combination with the Spectral Support Rule for pseudo-differential operators and pointwise estimates in terms of maximal functions of Peetre--Fefferman--Stein type.