The index of weighted singular integral operators with shifts and slowly oscillating data

Oleksiy Karlovych, Yuri I. Karlovich, Amarino B. Lebre

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4 Citations (Scopus)

Abstract

Let α and β be orientation-preserving diffeomorphism (shifts) of R+=(0,∞) onto itself with the only fixed points 0 and ∞. We establish a Fredholm criterion and calculate the index of the weighted singular integral operator with shifts (aI−bUα)Pγ ++(cI−dUβ)Pγ , acting on the space Lp(R+), where Pγ ±=(I±Sγ)/2 are the operators associated to the weighted Cauchy singular integral operator Sγ given by (Sγf)(t)=1πi∫R+(tτ)γf(τ)τ−tdτ with γ∈C satisfying 0<1/p+ℜγ<1, and Uα,Uβ are the isometric shift operators given by Uαf=(α)1/p(f∘α),Uβf=(β)1/p(f∘β), under the assumptions that the coefficients a,b,c,d and the derivatives α of the shifts are bounded and continuous on R+ and admit discontinuities of slowly oscillating type at 0 and ∞.

Original languageEnglish
Pages (from-to)606-630
Number of pages25
JournalJournal of Mathematical Analysis and Applications
Volume450
Issue number1
DOIs
Publication statusPublished - 1 Jun 2017

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Keywords

  • Fredholmness
  • Index
  • Orientation-preserving shift
  • Semi-almost periodic function
  • Slowly oscillating function
  • Weighted Cauchy singular integral operator

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