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

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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

Cite this

Karlovich, A. Y. (2015). Maximally Modulated Singular Integral Operators and their Applications to Pseudodifferential Operators on Banach Function Spaces. In K. Jarosz (Ed.), FUNCTION SPACES IN ANALYSIS (pp. 165-178). (Contemporary Mathematics; Vol. 645). AMER MATHEMATICAL SOC. https://doi.org/10.1090/conm/645/1.2908