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Maximally modulated singular integral operators and their applications to pseudodifferential operators on Banach function spaces

Citation:
Karlovich, Alexei Yu. "Maximally modulated singular integral operators and their applications to pseudodifferential operators on Banach function spaces." Function Spaces in Analysis. Contemporary Mathematics, 645. Ed. Krzysztof Jarosz. Providence, Rhode Island: American Mathematical Society, 2015. 165-178.

Abstract:

We prove that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space X(Rn) and on its associate space X(Rn) and a maximally modulated Calderón-Zygmund singular integral operator TΦ is of weak type (r,r) for all r(1,), then TΦ extends to a bounded operator on X(Rn). This theorem implies the boundedness of the maximally modulated Hilbert transform on variable Lebesgue spaces Lp()(R) under natural assumptions on the variable exponent p:R(1,). Applications of the above result to the boundedness and compactness of pseudodifferential operators with L(R,V(R))-symbols on variable Lebesgue spaces Lp()(R) are considered. Here the Banach algebra L(R,V(R)) consists of all bounded measurable V(R)-valued functions on R where V(R) is the Banach algebra of all functions of bounded total variation.

Related External Link

Preprint in arXiv:

http://arxiv.org/abs/1408.4400

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