@inbook {7433,
	title = {Maximally modulated singular integral operators and their applications to pseudodifferential operators on Banach function spaces},
	booktitle = {Function Spaces in Analysis. Contemporary Mathematics, 645},
	year = {2015},
	pages = {165-178},
	publisher = {American Mathematical Society},
	organization = {American Mathematical Society},
	address = {Providence, Rhode Island},
	abstract = {<script src={\textquoteright}https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML{\textquoteright}></script><p>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 \(X{\textquoteright}(\mathbb{R}^n)\) and a maximally modulated Calder{\'o}n-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.</p>
},
	url = {http://www.ams.org/books/conm/645/12908/conm645-12908.pdf},
	author = {Karlovich, Alexei Yu.},
	editor = {Krzysztof Jarosz}
}
