<?xml version="1.0" encoding="UTF-8"?><xml><records><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>5</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author><author><style face="normal" font="default" size="100%">Spitkovsky, Ilya M.</style></author></authors><secondary-authors><author><style face="normal" font="default" size="100%">Karlovich, Yuri I.</style></author><author><style face="normal" font="default" size="100%">Luigi Rodino</style></author><author><style face="normal" font="default" size="100%">Bernd Silbermann</style></author><author><style face="normal" font="default" size="100%">Spitkovsky, Ilya M.</style></author></secondary-authors></contributors><titles><title><style face="normal" font="default" size="100%">Pseudodifferential operators on variable Lebesgue spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Operator Theory, Pseudo-Differential Equations, and Mathematical Physics. Operator Theory: Advances and Applications, 228</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2013</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://link.springer.com/chapter/10.1007/978-3-0348-0537-7_9</style></url></web-urls></urls><publisher><style face="normal" font="default" size="100%">Birkhäuser</style></publisher><pub-location><style face="normal" font="default" size="100%">Basel</style></pub-location><pages><style face="normal" font="default" size="100%">173-183</style></pages><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;Let \(\mathcal{M}(\mathbb{R}^n)\) be the class of bounded away from one and infinity functions \(p:\mathbb{R}^n\to[1,\infty]\) 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 Hörmander class \(S_{\rho,\delta}^{n(\rho-1)}\) with \(0&amp;lt;\rho\le 1\), \(0\le\delta&amp;lt;1\), then the pseudodifferential operator \(\operatorname{Op}(a)\) is bounded on the variable Lebesgue space \(L^{p(\cdot)}(\mathbb{R}^n)\) 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 \(1/p(x)=\theta/p_0+(1-\theta)/p_1(x)\) where \(p_0\in(1,\infty)\), \(\theta\in(0,1)\), and \(p_1\in\mathcal{M}(\mathbb{R}^n)\). We prove that if \(a\in S_{1,0}^0\) slowly oscillates at infinity in the first variable, then the condition \[ \lim_{R\to\infty}\inf_{|x|+|\xi|\ge R}|a(x,\xi)|&amp;gt;0 \] is sufficient for the Fredholmness of \(\operatorname{Op}(a)\) on \(L^{p(\cdot)}(\mathbb{R}^n)\) whenever \(p\in\mathcal{M}^*(\mathbb{R}^n)\). Both theorems generalize pioneering results by Rabinovich and Samko [RS08] obtained for globally log-Hölder continuous exponents \(p\), constituting a proper subset of \(\mathcal{M}^*(\mathbb{R}^n)\). &lt;/p&gt;
</style></abstract></record></records></xml>