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

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.