Mapping properties of operator-valued pseudo-differential operators

TL;DR

This paper studies the mapping properties of operator-valued pseudo-differential operators using atomic decompositions of Triebel-Lizorkin spaces, establishing regularity results for various symbols on Euclidean and quantum tori.

Contribution

It introduces new regularity results for operator-valued pseudo-differential operators based on atomic decompositions of Triebel-Lizorkin spaces, extending to quantum tori.

Findings

Established $F_1^{ ext{alpha},c}$-regularity for regular symbols.

Proved $F_1^{ ext{alpha},c}$-regularity for forbidden symbols when $ ext{alpha}>0$.

Extended results to quantum tori.

Abstract

In this paper, we investigate the mapping properties of pseudo-differential operators with operator-valued symbols. Thanks to the smooth atomic decomposition of the operator-valued Triebel-Lizorkin spaces $F_1^{\alpha,c}(\mathbb{R}^d,\mathcal{M})$ obtained in our previous paper, we establish the $F_1^{\alpha,c}$-regularity of regular symbols for every $\alpha\in \mathbb{R}$, and the $F_1^{\alpha,c}$-regularity of forbidden symbols for $\alpha>0$. As applications, we obtain the same results on the usual and quantum tori.