PSEUDODIFFERENTIAL OPERATORS ON MIXED-NORM α-MODULATION SPACES

Abstract

Mixed-norm $\alpha$-modulation spaces were introduced recently by Cleanthous and Georgiadis (Trans. Amer. Math. Soc. 373:5 (2020), 3323–3356). The mixed-norm $\alpha$-modulation spaces $M_{\overrightarrow{p},q}^{s,\alpha }({ℝ}^n)$, $\alpha \in [0,1]$, form a family of smoothness spaces that contains the mixed-norm Besov spaces as special cases. We prove that a pseudodifferential operator $\sigma (x,D)$ with symbol in the Hörmander class $S_{\rho }^b$ extends to a bounded operator $\sigma (x,D):M_{\overrightarrow{p},q}^{s,\alpha }({ℝ}^n)\to M_{\overrightarrow{p},q}^{s-b,\alpha }({ℝ}^n)$, provided , $\overrightarrow{p}\in {(0,\infty )}^n$, and . This extends the known result that pseudodifferential operators, with symbol in the class $S_1^b$, maps the mixed-norm Besov space $B_{\overrightarrow{p},q}^s({ℝ}^n)$ into $B_{\overrightarrow{p},q}^{s-b}({ℝ}^n)$.