Fredholmness of Nonlocal Singular Integral Operators with Slowly Oscillating Data

Let p ∈ (1, ∞), α be an orientation-preserving homeomorphism of [0, ∞] onto itself with only two fixed points 0 and ∞, whose restriction to ℝ + = ( 0 , ∞ ) $$\mathbb{R}_{+} = (0,\infty )$$ is a diffeomorphism, and let U α be the isometric shift operator acting on the Lebesgue space L p ( ℝ + ) $$L^{p}(\mathbb{R}_{+})$$ 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 ( ℝ + ) $$L^{p}(\mathbb{R}_{+})$$ , where P ± = ( I ± S ℝ + ) ∕ 2 $$P_{\pm } = (I \pm S_{\mathbb{R}_{+}})/2$$ , I is the identity operator, S ℝ + $$S_{\mathbb{R}_{+}}$$ is the Cauchy singular integral operator over ℝ + $$\mathbb{R}_{+}$$ , A ± are functional operators of the form A ± = ∑ k ∈ ℤ a k ± U α k , where ∥ A ± ∥ W = ∑ k ∈ ℤ ∥ a k ± ∥ L ∞ ( ℝ + )