Fredholmness of Nonlocal Singular Integral Operators with Slowly Oscillating Data

Abstract

Let p ∈ (1, ∞), α be an orientation-preserving homeomorphism of [0, ∞] onto itself with only two fixed points 0 and ∞, whose restriction to $\mathbb{R}_{+} = (0,\infty )$ is a diffeomorphism, and let U _α be the isometric shift operator acting on the Lebesgue space $L^{p}(\mathbb{R}_{+})$ by the rule U _α f = (α ^′)^1∕p(f ∘α). We establish sufficient conditions for the Fredholmness of the nonlocal singular integral operator N = A ₊ P ₊ + A ₋ P ₋ on the space $L^{p}(\mathbb{R}_{+})$, where $P_{\pm } = (I \pm S_{\mathbb{R}_{+}})/2$, I is the identity operator, $S_{\mathbb{R}_{+}}$ is the Cauchy singular integral operator over $\mathbb{R}_{+}$, A _± are functional operators of the form

$$\displaystyle{A_{\pm } =\sum _{k\in \mathbb{Z}}a_{k}^{\pm }U_{\alpha }^{k},\ \ \mbox{ where }\ \|A_{ \pm }\|_{W} =\sum _{k\in \mathbb{Z}}\|a_{k}^{\pm }\|_{ L^{\infty }(\mathbb{R}_{+})} <\infty,}$$

under assumptions that the functions logα ^′ and coefficients a _k ^± for all $k \in \mathbb{Z}$ are bounded and continuous on $\mathbb{R}_{+}$ and may have slowly oscillating discontinuities at 0 and ∞.