论文标题
混合声 - $α$调制空间的假差异操作员
Pseudodifferential operators on Mixed-Norm $α$-modulation spaces
论文作者
论文摘要
Cleanthous和Georgiadis [Trans。\Amer。\Math。\ Soc。\ Soc。\ 373（2020），否。 5，3323-3356]。混合-norm空间$ m^{s，α} _ {\ vec {p}，q}（\ mathbb {r}^n）$，$α\ in [0,1] $，形成一个平滑度的家族，其中包含混合norm besov空间（特殊情况）。在本文中，我们证明了hörmander类$ s $ s^b_ρ$在hörmander类$ s $ s^b_ρ$中的伪数字运算符$σ（x，d）$扩展到有界的操作员$σ（x，x，d）\ colon m^{s，α}，α} _ {\ vec {p}，q}，q}，\ n） m^{s-b，α} _ {\ vec {p}，q}（\ mathbb {r}^n）$提供$ 0 <α\ leqρ\ leq 1 $， $ \ vec {p} \在（0，\ infty）^n $和$ 0 <q <\ infty $。结果扩展了已知的结果，即在类$ s^b_ {1} $中具有符号的伪数运算符映射混合 - norm besov space $ b^s _ {\ vec {p}，q}，\ mathbb {r}^n）$ in $ b^{s-b} _ {\ vec {p}，q}（\ mathbb {r}^n）$。
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 $B^{s-b}_{\vec{p},q}(\mathbb{R}^n)$.
