Pseudodifferential Operators on Variable Lebesgue Spaces

Let $\mathcal{M}({\mathbb{R}}^n)$ be the class of bounded away from one and infinity functions $p:{\mathbb{R}}^n\to [1,\mathrm{\infty }]$ such that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space $L^{p(\cdot )}({\mathbb{R}}^n)$. We show that if $a$ belongs to the H"ormander class $S_{\rho ,\delta }^{n(\rho -1)}$ with $0<\rho \le 1$, $0\le \delta <1$, then the pseudodifferential operator $\text{Op}(a)$ is bounded on the variable Lebesgue space $L^{p(\cdot )}({\mathbb{R}}^n)$ provided that $p\in \text{cM}({\mathbb{R}}^n)$. Let ${\mathcal{M}}^{\ast }({\mathbb{R}}^n)$ be the class of variable exponents $p\in \mathcal{M}({\mathbb{R}}^n)$ represented as $1\mathrm{/}p(x)=\theta \mathrm{/}{p}_0+(1-\theta )\mathrm{/}{p}_1(x)$ where $p_0\in (1,\mathrm{\infty })$, $\theta \in (0,1)$, and $p_1\in \mathcal{M}({\mathbb{R}}^n)$. We prove that if $a\in S_{1,0}^0$ slowly oscillates at infinity in the first variable, then the condition $$\underset{R\to \mathrm{\infty }}{\lim }\underset{\mathrm{\mid }x\mathrm{\mid }+\mathrm{\mid }\xi \mathrm{\mid }\ge R}{\inf }\mathrm{\mid }a(x,\xi )\mathrm{\mid }>0$$ is sufficient for the Fredholmness of $\text{Op}(a)$ on $L^{p(\cdot )}({\mathbb{R}}^n)$ whenever $p\in {\text{cM}}^{\ast }({\mathbb{R}}^n)$. Both theorems generalize pioneering results by Rabinovich and Samko \cite{RS08} obtained for globally log-H"older continuous exponents $p$, constituting a proper subset of ${\mathcal{M}}^{\ast }({\mathbb{R}}^n)$.