Fundamental Results for Pseudo-Differential Operators of Type $\mathbf{1},\mathbf{1}$

TL;DR

This paper advances the theory of type 1,1 pseudo-differential operators by establishing their continuity on temperate distributions under certain conditions and developing a paradifferential decomposition to analyze their structure.

Contribution

It introduces a systematic framework for type 1,1 operators, including their domain properties and a paradifferential decomposition, extending previous definitions and results.

Findings

Type 1,1 operators are continuous on temperate distributions under the twisted diagonal condition.

Paradifferential decomposition is derived for type 1,1 operators.

Symmetric terms cause domain restrictions, while other terms satisfy the twisted diagonal condition.

Abstract

This paper develops some deeper consequences of an extended definition, proposed previously by the author, of pseudo-differential operators that are of type $1,1$ in H\"ormander's sense. Thus, it contributes to the long-standing problem of creating a systematic theory of such operators. It is shown that type $1,1$-operators are defined and continuous on the full space of temperate distributions, if they fulfil H\"ormander's twisted diagonal condition, or more generally if they belong to the self-adjoint subclass; and that they are always defined on the temperate smooth functions. As a main tool the paradifferential decomposition is derived for type $1,1$-operators, and to confirm a natural hypothesis the symmetric term is shown to cause the domain restrictions; whereas the other terms are shown to define nice type $1,1$-operators fulfilling the twisted diagonal condition. The decomposition is analysed in the type $1,1$-context by combining the Spectral Support Rule and the factorisation inequality, which gives pointwise estimates of pseudo-differential operators in terms of maximal functions.