Pseudodifferential operators on variable Lebesgue spaces

Let $\mathcal{M}(\mathbb{R}^n)$ be the class of bounded away fromone and infinity functions $p:\mathbb{R}^n\to[1,\infty]$ such thatthe Hardy-Littlewood maximal operator is bounded on the variableLebesgue 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 pseudodifferentialoperator $\Op(a)$ is bounded on the variable Lebesgue space$L^{p(\cdot)}(\R^n)$ provided that $p\in\cM(\R^n)$. Let$\mathcal{M}^*(\mathbb{R}^n)$ be the class of variable exponents$p\in\mathcal{M}(\mathbb{R}^n)$ represented as$1/p(x)=\theta/p_0+(1-\theta)/p_1(x)$ where $p_0\in(1,\infty)$,$\theta\in(0,1)$, and $p_1\in\mathcal{M}(\mathbb{R}^n)$. We provethat if $a\in S_{1,0}^0$ slowly oscillates at infinity in thefirst variable, then the condition\[\lim_{R\to\infty}\inf_{|x|+|\xi|\ge R}|a(x,\xi)|>0\]is sufficient for the Fredholmness of $\Op(a)$ on $L^{p(\cdot)}(\R^n)$whenever $p\in\cM^*(\R^n)$. Both theorems generalize pioneeringresults by Rabinovich and Samko \cite{RS08} obtained for globallylog-H\"older continuous exponents $p$, constituting a proper subsetof $\mathcal{M}^*(\mathbb{R}^n)$.