  The main result is that every pseudo-differential operator of type 1,1 and
order $d$ is continuous from the Triebel--Lizorkin space $F^d_{p,1}$ to $L_p$,
$1\le p<\infty$, and that this is optimal within the Besov and
Triebel--Lizorkin scales.The proof also leads to the known continuity for
$s>d$, while for all real $s$ the sufficiency of H\"ormander's condition on the
twisted diagonal is carried over to the Besov and Triebel--Lizorkin framework.
To obtain this, type 1,1-operators are extended to distributions with compact
spectrum, and Fourier transformed operators of this type are on such
distributions proved to satisfy a support rule, generalising the rule for
convolutions. Thereby the use of reduced symbols, as introduced by Coifman and
Meyer, is replaced by direct application of the paradifferential methods. A few
flaws in the literature have been detected and corrected.
