Necessary Fredholm conditions for weighted singular integral operators with shifts and slowly oscillating data

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

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

We extend the main result of [12] to the case of more general weighted singular integral operators with two shifts of the form (aI - bUα)Pγ + + (cI - dUβ)Pγ -, acting on the space Lp(ℝ+), 1 < p < ∞, where Pγ ± = (I ± Sγ)/2 are operators associated with the weighted Cauchy singular integral operator Sγ, given by with γ εℂ satisfying 0 < 1/p + γ < 1, and Uα, Uβ are the isometric shift operators given by Uαf = (α')1/p(f ο α), Uβf = (β')1/p(f ο β), generated by diffeomorphisms α β of ℝ+ onto itself having only two βxed points at the endpoints 0 and ∞, under the assumptions that the coefficients a; b; c; d and the derivatives α', β' of the shifts are bounded and continuous on ℝ+ and admit discontinuities of slowly oscillating type at 0 and ∞.

Original languageEnglish
Pages (from-to)365-399
Number of pages35
JournalJournal of Integral Equations and Applications
Volume29
Issue number3
DOIs
Publication statusPublished - 2017

Keywords

  • Fredholmness
  • Orientation-preserving shift
  • Slowly oscillating function
  • Weighted Cauchy singular integral operator

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