Pseudodifferential Operators on Variable Lebesgue Spaces

Abstract

Let $\mathcal{M}(\mathbb{R}^n)$be the class of bounded away from one and infinity functions $p : \mathbb{R}^n \rightarrow [1,\infty]$ such that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space $L^{p(.)}(\mathbb{R}^{(n)}$. We show that if a belongs to the Hörmander class $S ^{n(\rho-1)}_{\rho\delta}$ with $0 < \rho \leq 1, 0 \leq \delta < 1$ 1, then the pseudodifferential operator Opa is bounded on $L^{p(.)}{(\mathbb{R}^n)}$ provided that $p\in \mathcal{M}{(\mathbb{R}^n)}$. Let $\mathcal{M}*{(\mathbb{R}^n)}$ be the class of variable exponents $p\in \mathcal{M}{(\mathbb{R}^n)}$ represented as ${\raise0.7ex\hbox{$1$} \!\mathord{\left/{\vphantom {1 {p(x)}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${p(x)}$}}$ ${\raise0.7ex\hbox{$\theta $} \!\mathord{\left/{\vphantom {\theta {po}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${po}$}} + {\raise0.7ex\hbox{${(1 - 0)}$} \!\mathord{\left/{\vphantom {{(1 -0)}{p1}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${p1}$}}(x) {\rm where \, po} \in (1,\infty),(\theta)\in (0,1) \rm {and}p\in \mathcal{M}{(\mathbb{R}^n)}.\rm {We prove that if} a \in s^{0}_{1,0}$ ${\lim}\kern-\nulldelimiterspace\!\lower0.7ex\hbox{${po}$} + {\raise0.7ex\hbox{${(1 - 0)}$} \!\mathord{\left/{\vphantom {{(1 - 0)} {p1}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${p1}$}}(x) {\rm where \, po} \in (1,\infty),(\theta)\in (0,1) \rm {and} p\in \mathcal{M}{(\mathbb{R}^n)}.\rm {We prove that if} a \in s^{0}_{1,0}$ slowly oscillates at infinity in the first variable, then the condition $\lim_{R} \rightarrow \infty \rm{inf}_{|x|+|\xi|\geq|R}{a(x,\xi|)}>0$ is sufficient for the Fredholmness of Op(A) on $p\in \mathcal{M}{(\mathbb{R}^n)}$ $L^{p(.)}{(\mathbb{R}^n)}$ whenever Opa $L^{p(.)}{(\mathbb{R}^n)}$ Both theorems generalize pioneering results by Rabinovich and Samko [24] obtained for globally log-Hölder continuous exponents p constituting a proper subset of $p\in \mathcal{M}{(\mathbb{R}^n)}$

To Professor Vladimir Rabinovich on the occasion of his 70th birthday

Mathematics Subject Classification (2010). Primary 47G30; Secondary 42B25, 46E30.