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

Abstract

This is a continuation of recent work on the general definition of pseudo-differential operators of type $1,1$, in Hörmander's sense. Continuity in $L_p$-Sobolev spaces and Hölder--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.

18 pages. The accepted manuscript. (Final version appeared on 14 September 2012 in Mathematische Nachrichten.)

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• On the trace problem for Triebel--Lizorkin spaces with mixed norms
• A direct proof of Sobolev embeddings for quasi-homogeneous Lizorkin--Triebel spaces with mixed norms
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Cited by in corpus (1)

• Fundamental Results for Pseudo-Differential Operators of Type $1,1$