On the Theory of Type 1,1-Operators

TL;DR

This dissertation provides a comprehensive systematic theory of type 1,1-operators, proving their pseudo-locality, analyzing their support and spectrum effects, and establishing boundedness in various function spaces.

Contribution

It introduces a general definition of type 1,1-operators, proves their pseudo-locality conjecture, and extends the analysis to their support, spectrum, and boundedness properties in advanced function spaces.

Findings

Type 1,1-operators are pseudo-local.

Operators can alter support and spectrum of functions.

Operators are bounded in Sobolev, Besov, and Lizorkin--Triebel spaces.

Abstract

This dissertation concerns the pseudo-differential operators of type 1,1. These have been known especially since around 1980, when it was shown that they play an important role in the treatment of fully non-linear partial differential equations. First an account of the historical development in the area is given, including fundamental contributions due to G. Bourdaud and L. H\"ormander in 1988-89, with concise remarks on the authors contributions. Secondly a detailed exposition is given of the systematic theory of type 1,1-operators, based on the general definition of such operators proposed by the author in 2008. This includes an account of how the previous extensions are generalised hereby. Moreover, the conjecture from 1978 by C. Parenti and L. Rodino that type 1,1-operators are pseudo-local is proved in this framework. It is also analysed how such operators can change the support and the spectrum of the functions they act on. Furthermore, the question of which temperate distributions such operators have in their domains have been extensively treated, and departing from H\"ormander's analysis of the operators fulfilling the twisted diagonal condition or, more generally, belong to the self-adjoint subclass, such operators are shown to be everywhere defined. Finally the paradifferential splittings and the resulting boundedness in Lp Sobolev spaces, and the more general Besov and Lizorkin--Triebel spaces, are amply discussed.