Maximally Modulated Singular Integral Operators and their Applications to Pseudodifferential Operators on Banach Function Spaces

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Abstract

We prove that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space X(R-n) and on its associate space X'(R-n) and a maximally modulated Calderon-Zygmund singular integral operator T-Phi is of weak type (r, r) for all r is an element of E (1, infinity), then T-Phi extends to a bounded operator on X(R-n). This theorem implies the boundedness of the maximally modulated Hilbert transform on variable Lebesgue spaces L-p((.)) (H) under natural assumptions on the variable exponent p : R -> (1, infinity). Applications of the above result to the boundedness and compactness of pseudodifferential operators with L-infinity, V(R))-symbols on variable Lebesgue spaces L-p((.)) (H) are considered. Here the Banach algebra L-infinity(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.

Original languageEnglish
Title of host publicationFUNCTION SPACES IN ANALYSIS
EditorsK Jarosz
PublisherAMER MATHEMATICAL SOC
Pages165-178
Number of pages14
ISBN (Print)978-1-4704-1694-2
DOIs
Publication statusPublished - 2015
Event7th Conference on Function Spaces - Edwardsville, Israel
Duration: 20 May 201424 May 2014

Publication series

NameContemporary Mathematics
PublisherAMER MATHEMATICAL SOC
Volume645
ISSN (Print)0271-4132

Conference

Conference7th Conference on Function Spaces
CountryIsrael
CityEdwardsville
Period20/05/1424/05/14

Keywords

  • Maximally modulated singular integral operator
  • Calderon-Zygmund operator
  • Hilbert transform
  • pseudodifferential operator with non-regular symbol
  • Banach function space
  • variable Lebesgue space
  • LEBESGUE SPACES
  • BOUNDEDNESS
  • COMPACTNESS
  • SYMBOLS

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