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Nonlocal Singular Integral Operators with Slowly Oscillating Data

  • Yuri I. Karlovich
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 181)

Abstract

The paper is devoted to studying the Fredholmness of (nonlocal) singular integral operators with shifts N = (Σ a g + V g )P + + (Σa g V g )P on weighted Lebesgue spaces L p (Γ,w) where 1 < p < ∞, Γ is an unbounded slowly oscillating Carleson curve, w is a slowly oscillating Muckenhoupt weight, the operators P ± = 1/2 (I ± S Γ) are related to the Cauchy singular integral operator SΓ, a g ± are slowly oscillating coefficients, V g are shift operators given by V g f = f o g, and g are slowly oscillating shifts in a finite subset of a subexponential group G acting topologically freely on Γ. The Fredholm criterion for N consists of two parts: of an invertibility criterion for polynomial functional operators A ± = Σa g ± V g in terms of invertibility of corresponding discrete operators on the space l p (G), and of a condition of local Fredholmness of N at the endpoints of Γ established by applying Mellin pseudodifferential operators with compound slowly oscillating V (ℝ)-valued symbols where V (ℝ) is the Banach algebra of absolutely continuous functions of bounded total variation on ℝ.

Keywords

Singular integral operator with shifts slowly oscillating data Fredholmness weighted Lebesgue space Mellin pseudodifferential operator compound symbol 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Yuri I. Karlovich
    • 1
  1. 1.Facultad de CienciasUniversidad Autónoma del Estado de MorelosCuernavaca, MorelosMéxico

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