Sharp estimates for pseudo-differential operators of type (1,1) on Triebel-Lizorkin and Besov spaces

TL;DR

This paper establishes sharp boundedness estimates for type (1,1) pseudo-differential operators on Triebel-Lizorkin and Besov spaces, extending previous results to include the case p=∞ and exploring conditions for boundedness.

Contribution

The work extends boundedness results of pseudo-differential operators to the case p=∞ and analyzes their mapping properties on advanced function spaces.

Findings

Operators are bounded from $F_p^{s+m,q}$ to $F_p^{s,q}$ under certain conditions.

Extension of $F$-boundedness to the case p=∞.

Operators map $F_{ ext{infinity}}^{m,1}$ into BMO when s=0.

Abstract

Pseudo-differential operators of type $(1,1)$ and order $m$ are continuous from $F_p^{s+m,q}$ to $F_p^{s,q}$ if $s>d/\min{(1,p,q)}-d$ for $0<p<\infty$, and from $B_p^{s+m,q}$ to $B_{p}^{s,q}$ if $s>d/\min{(1,p)}-d$ for $0<p\leq\infty$. In this work we extend the $F$-boundedness result to $p=\infty$. Additionally, we prove that the operators map $F_{\infty}^{m,1}$ into $bmo$ when $s=0$, and consider H\"ormander's twisted diagonal condition for arbitrary $s\in\mathbb{R}$. We also prove that the restrictions on $s$ are necessary conditions for the boundedness to hold.