On a Weighted Singular Integral Operator with Shifts and Slowly Oscillating Data

Abstract

Let $\alpha ,\beta$ be orientation-preserving diffeomorphism (shifts) of ${\mathbb{R}}_{+}=(0,\infty )$ onto itself with the only fixed points $0$ and $\infty$ and $U_{\alpha },U_{\beta }$ be the isometric shift operators on $L^p({\mathbb{R}}_{+})$ given by $U_{\alpha }f=({\alpha }^{\prime }{)}^{1/p}(f\circ \alpha )$, $U_{\beta }f=({\beta }^{\prime }{)}^{1/p}(f\circ \beta )$, and $P_2^{\pm }=(I\pm S_2)/2$ where \[ (S_2 f)(t):=\frac{1}{πi}\int\limits_0^\infty \left(\frac{t}τ\right)^{1/2-1/p}\frac{f(τ)}{τ-t}\,dτ, \quad t\in\mathbb{R}_+, \] is the weighted Cauchy singular integral operator. We prove that if ${\alpha }^{\prime },{\beta }^{\prime }$ and $c,d$ are continuous on ${\mathbb{R}}_{+}$ and slowly oscillating at $0$ and $\infty$, and \[ \limsup_{t\to s}|c(t)|<1, \quad \limsup_{t\to s}|d(t)|<1, \quad s\in\{0,\infty\}, \] then the operator $(I-c{U}_{\alpha })P_2^{+}+(I-d{U}_{\beta })P_2^{-}$ is Fredholm on $L^p({\mathbb{R}}_{+})$ and its index is equal to zero. Moreover, its regularizers are described.

28 pages. arXiv admin note: text overlap with arXiv:1405.0368