Type 1,1-operators on spaces of temperate distributions

This paper is a follow-up on the author’s general definition of pseudo-differential
operators of type 1,1, in Hörmander’s sense. It is shown that such operators are always defined on the smooth functions that are temperate; and moreover are defined and continuous on the space of temperate distributions, whenever they fulfil the twisted diagonal condition of Hörmander, or more generally when they belong to the self-adjoint subclass. Continuity in L_p -Sobolev spaces and Hölder–Zygmund spaces, and more generally in Besov and Lizorkin–Triebel spaces, is for
positive smoothness also proved on the basis of the definition. These continuity results are extended to arbitrary real smoothness indices for operators that fulfil the twisted diagonal condition or belong to the self-adjoint subclass. With systematic Littlewood–Paley analysis the well-known paradifferential decomposition is also derived for type 1,1-operators. The proofs are based on a
spectral support rule for pseudo-differential operators in combination with pointwise estimates in terms of maximal functions.