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

## TL;DR

This paper proves the boundedness of type 1,1 pseudo-differential operators from Triebel--Lizorkin spaces to Lebesgue spaces, extending classical conditions and establishing optimal domain results.

## Contribution

It establishes the continuity of type 1,1 pseudo-differential operators on Triebel--Lizorkin spaces and extends H"ormander's twisted diagonal condition within this framework.

## Key findings

- Operators are continuous from $F^d_{p,1}$ to $L_p$ for $1 extless p extless \infty$.
- This is the largest possible domain for such operators among Besov and Triebel--Lizorkin spaces.
- Extension of H"ormander's condition to this setting using a support rule.

## Abstract

Pseudo-differential operators of type 1,1 are proved continuous from the Triebel--Lizorkin space $F^d_{p,1}$ to $L_p$ for $1\le p<\infty$, 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.