# Nonlocal Singular Integral Operators with Slowly Oscillating Data

## Abstract

The paper is devoted to studying the Fredholmness of (nonlocal) singular integral operators with shifts *N* = (Σ *a* _{g} ^{+} *V* _{ g })*P* _{+} + (Σ*a* _{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 *S*Γ, *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* _{±} = Σ*a* _{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## Preview

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