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

Mixed-norm $α$-modulation spaces were introduced recently by Cleanthous and Georgiadis [Trans.\ Amer.\ Math.\ Soc.\ 373 (2020), no. 5, 3323-3356]. The mixed-norm spaces $M^{s,α}_{\vec{p},q}(\mathbb{R}^n)$, $α\in [0,1]$, 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 $σ(x,D)$ with symbol in the Hörmander class $S^b_ρ$ extends to a bounded operator $σ(x,D)\colon M^{s,α}_{\vec{p},q}(\mathbb{R}^n) \rightarrow M^{s-b,α}_{\vec{p},q}(\mathbb{R}^n)$ provided $0<α\leq ρ\leq 1$, $\vec{p}\in (0,\infty)^n$, 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 $B^{s-b}_{\vec{p},q}(\mathbb{R}^n)$.

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