  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.
