Lp-theory of type 1, 1-operators


  • Dedicated to Professor Hans Triebel on the occasion of his seventy-fifth birthday


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 Lp-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.