Domains of type 1,1 operators: a case for Triebel--Lizorkin spaces

Pseudo-differential operators of type 1,1 are proved continuous from the Triebel–Lizorkin space $F_{p,1}^d$ to $L_p$ for $1\le p<\mathrm{\infty }$, when of order d, and this is the largest possible domain among the Besov and Triebel–Lizorkin spaces. H"ormander’s condition on the twisted diagonal is extended to this framework, using a general support rule for Fourier transformed pseudo-differential operators.