Nonlocal Singular Integral Operators with Slowly Oscillating Data
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.
KeywordsSingular integral operator with shifts slowly oscillating data Fredholmness weighted Lebesgue space Mellin pseudodifferential operator compound symbol
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