# Pseudodifferential operators on Mixed-Norm $α$-modulation spaces

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Publikation: Working paper/PreprintPreprint

## Abstrakt

Mixed-norm $\alpha$-modulation spaces were introduced recently by Cleanthous and Georgiadis [Trans.\ Amer.\ Math.\ Soc.\ 373 (2020), no. 5, 3323-3356]. The mixed-norm spaces $M^{s,\alpha}_{\vec{p},q}(\mathbb{R}^n)$, $\alpha\in [0,1]$, form a family of smoothness spaces that contain the mixed-norm Besov spaces as special cases. In this paper we prove that a pseudodifferential operator $\sigma(x,D)$ with symbol in the H\"ormander class $S^b_{\rho}$ extends to a bounded operator $\sigma(x,D)\colon M^{s,\alpha}_{\vec{p},q}(\mathbb{R}^n) \rightarrow M^{s-b,\alpha}_{\vec{p},q}(\mathbb{R}^n)$ provided $0<\alpha\leq \rho\leq 1$, $\vec{p}\in (0,\infty)^n$, and \$0
Originalsprog Engelsk Udgivet - 29 mar. 2022

## Emneord

• math.FA
• Primary 47G30, 46E35, Secondary 47B38