Sufficient Conditions for Fredholmness of Singular Integral Operators with Shifts and Slowly Oscillating Data

Abstract

Suppose α is an orientation preserving diffeomorphism (shift) of $${{\mathbb{R}}_+=(0,\infty)}$$ onto itself with the only fixed points 0 and ∞. We establish sufficient conditions for the Fredholmness of the singular integral operator with shift $$(aI-bW_\alpha)P_++(cI-dW_\alpha)P_-$$ acting on $${L^p({\mathbb{R}}_+)}$$ with 1 < p < ∞, where P ± = (I ± S)/2, S is the Cauchy singular integral operator, and $${{{W_{\alpha}f=f\circ\alpha}}}$$ is the shift operator, under the assumptions that the coefficients a, b, c, d and the derivative α′ of the shift are bounded and continuous on $${{\mathbb{R}}_+}$$ and may admit discontinuities of slowly oscillating type at 0 and ∞.