Pseudodifferential operators on variable lebesgue spaces

Alexei Yu Karlovich, Ilya M. Spitkovsky

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

Abstract

Let M(ℝn) be the class of bounded away from one and infinity functions p: ℝn → [1,∞] such that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space Lp(⋅)(ℝn). We show that if a belongs to the Hörmander class Sn(ρ-1) ρ, δ with 0 < ρ ≤ 1, 0 ≤ δ < 1, then the pseudodifferential operator Op(a) is bounded on Lp(⋅)(ℝn) provided that p∈M(ℝn). Let M* (ℝn) be the class of variable exponents p∈M(ℝn) represented as 1/p(x) = θ/p0 + (1 – θ)/p1(x) where p0 ∈ (1,∞), θ ∈(0, 1), and p1 ∈M(ℝn). We prove that if a ∈ S0 1,0 slowly oscillates at infinity in the first variable, then the condition (Formula presented.) is sufficient for the Fredholmness of Op(a) on Lp(⋅)(ℝn) whenever p∈M* (ℝn). Both theorems generalize pioneering results by Rabinovich and Samko [24] obtained for globally log-Hölder continuous exponents p, constituting a proper subset of M* (ℝn).

Original languageEnglish (US)
Title of host publicationOperator Theory, Pseudo-Differential Equations, and Mathematical Physics
EditorsYuri I. Karlovich, Luigi Rodino, Bernd Silbermann, Ilya M. Spitkovsky
PublisherSpringer International Publishing
Pages173-183
Number of pages11
ISBN (Print)9783034805360
DOIs
StatePublished - 2013
EventInternational workshop on Analysis, Operator Theory, and Mathematical Physics, 2012 - Ixtapa, Mexico
Duration: Jan 23 2012Jan 27 2012

Publication series

NameOperator Theory: Advances and Applications
Volume228
ISSN (Print)0255-0156
ISSN (Electronic)2296-4878

Other

OtherInternational workshop on Analysis, Operator Theory, and Mathematical Physics, 2012
CountryMexico
CityIxtapa
Period1/23/121/27/12

Keywords

  • Fefferman-Stein sharp maximal operator
  • Fredholmness
  • Hardy-Littlewood maximal operator
  • Hörmander symbol
  • Pseudodifferential operator
  • Slowly oscillating symbol
  • Variable Lebesgue space

ASJC Scopus subject areas

  • Analysis

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