<?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></authors><secondary-authors><author><style face="normal" font="default" size="100%">Krzysztof Jarosz</style></author></secondary-authors></contributors><titles><title><style face="normal" font="default" size="100%">Maximally modulated singular integral operators and their applications to pseudodifferential operators on Banach function spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Function Spaces in Analysis. Contemporary Mathematics, 645</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2015</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://www.ams.org/books/conm/645/12908/conm645-12908.pdf</style></url></web-urls></urls><publisher><style face="normal" font="default" size="100%">American Mathematical Society</style></publisher><pub-location><style face="normal" font="default" size="100%">Providence, Rhode Island</style></pub-location><pages><style face="normal" font="default" size="100%">165-178</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;We prove that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space \(X(\mathbb{R}^n)\) and on its associate space \(X'(\mathbb{R}^n)\) and a maximally modulated Calderón-Zygmund singular integral operator \(T^{\Phi}\) is of weak type \((r,r)\) for all \(r\in(1,\infty)\), then \(T^{\Phi}\) extends to a bounded operator on \(X(\mathbb{R}^n)\). This theorem implies the boundedness of the maximally modulated Hilbert transform on variable Lebesgue spaces \(L^{p(\cdot)}(\mathbb{R})\) under natural assumptions on the variable exponent \(p:\mathbb{R}\to(1,\infty)\). Applications of the above result to the boundedness and compactness of pseudodifferential operators with \(L^\infty(\mathbb{R},V(\mathbb{R}))\)-symbols on variable Lebesgue spaces \(L^{p(\cdot)}(\mathbb{R})\) are considered. Here the Banach algebra \(L^\infty(\mathbb{R},V(\mathbb{R}))\) consists of all bounded measurable \(V(\mathbb{R})\)-valued functions on \(\mathbb{R}\) where \(V(\mathbb{R})\) is the Banach algebra of all functions of bounded total variation.&lt;/p&gt;
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