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

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 $M_{\vec{p},q}^{s,\alpha }({\mathbb{R}}^n)$, $\alpha \in [0,1]$, 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 Hörmander class $S_{\rho }^b$ extends to a bounded operator $\sigma (x,D):M_{\vec{p},q}^{s,\alpha }({\mathbb{R}}^n)\to M_{\vec{p},q}^{s-b,\alpha }({\mathbb{R}}^n)$ provided $0<\alpha \le \rho \le 1$, $\vec{p}\in (0,\infty {)}^n$, and $0<q<\infty$. The result extends the known result that pseudodifferential operators with symbol in the class $S_1^b$ maps the mixed-norm Besov space $B_{\vec{p},q}^s({\mathbb{R}}^n)$ into $B_{\vec{p},q}^{s-b}({\mathbb{R}}^n)$.