Pseudodifferential operators on Mixed-Norm $\alpha$-modulation spaces

TL;DR

This paper proves that pseudodifferential operators with symbols in Hörmander classes act boundedly on mixed-norm α-modulation spaces, extending known results from mixed-norm Besov spaces.

Contribution

It establishes boundedness of pseudodifferential operators on mixed-norm α-modulation spaces for symbols in Hörmander classes, generalizing previous results.

Findings

Boundedness of pseudodifferential operators on mixed-norm α-modulation spaces.

Extension of known results from Besov spaces to α-modulation spaces.

Applicable for symbols in Hörmander class $S^b_{\rho}$ with $0<\alpha\leq \rho\leq 1.

Abstract

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 $0<q<\infty$. The result extends the known result that pseudodifferential operators with symbol in the class $S^b_{1}$ maps the mixed-norm Besov space $B^s_{\vec{p},q}(\mathbb{R}^n)$ into $B^{s-b}_{\vec{p},q}(\mathbb{R}^n)$.