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

Research paper by Alexei Yu. Karlovich, Yuri I. Karlovich; Amarino B. Lebre

Indexed on: 11 Aug '16Published on: 01 Aug '16Published in: Complex Analysis and Operator Theory



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 \(U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha )\) , \(U_\beta f=(\beta ')^{1/p}(f\circ \beta )\) , and \(P_2^\pm =(I\pm S_2)/2\) where $$\begin{aligned} (S_2 f)(t):=\frac{1}{\pi i}\int \limits _0^\infty \left( \frac{t}{\tau }\right) ^{1/2-1/p}\frac{f(\tau )}{\tau -t}\,d\tau , \quad t\in \mathbb {R}_+, \end{aligned}$$ is the weighted Cauchy singular integral operator. We prove that if \(\alpha ',\beta '\) and \(c,d\) are continuous on \(\mathbb {R}_+\) and slowly oscillating at \(0\) and \(\infty \) , and $$\begin{aligned} \limsup _{t\rightarrow s} c(t) <1, \quad \limsup _{t\rightarrow s} d(t) <1, \quad s\in \{0,\infty \}, \end{aligned}$$ then the operator \((I-cU_\alpha )P_2^++(I-dU_\beta )P_2^-\) is Fredholm on \(L^p(\mathbb {R}_+)\) and its index is equal to zero. Moreover, its regularizers are described. 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 \(U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha )\) , \(U_\beta f=(\beta ')^{1/p}(f\circ \beta )\) , and \(P_2^\pm =(I\pm S_2)/2\) where $$\begin{aligned} (S_2 f)(t):=\frac{1}{\pi i}\int \limits _0^\infty \left( \frac{t}{\tau }\right) ^{1/2-1/p}\frac{f(\tau )}{\tau -t}\,d\tau , \quad t\in \mathbb {R}_+, \end{aligned}$$ is the weighted Cauchy singular integral operator. We prove that if \(\alpha ',\beta '\) and \(c,d\) are continuous on \(\mathbb {R}_+\) and slowly oscillating at \(0\) and \(\infty \) , and $$\begin{aligned} \limsup _{t\rightarrow s} c(t) <1, \quad \limsup _{t\rightarrow s} d(t) <1, \quad s\in \{0,\infty \}, \end{aligned}$$ then the operator \((I-cU_\alpha )P_2^++(I-dU_\beta )P_2^-\) is Fredholm on \(L^p(\mathbb {R}_+)\) and its index is equal to zero. Moreover, its regularizers are described. \(\alpha ,\beta \) \(\alpha ,\beta \) \(\mathbb {R}_+=(0,\infty )\) \(\mathbb {R}_+=(0,\infty )\) \(0\) \(0\) \(\infty \) \(\infty \) \(U_\alpha ,U_\beta \) \(U_\alpha ,U_\beta \) \(L^p(\mathbb {R}_+)\) \(L^p(\mathbb {R}_+)\) \(U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha )\) \(U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha )\) \(U_\beta f=(\beta ')^{1/p}(f\circ \beta )\) \(U_\beta f=(\beta ')^{1/p}(f\circ \beta )\) \(P_2^\pm =(I\pm S_2)/2\) \(P_2^\pm =(I\pm S_2)/2\) $$\begin{aligned} (S_2 f)(t):=\frac{1}{\pi i}\int \limits _0^\infty \left( \frac{t}{\tau }\right) ^{1/2-1/p}\frac{f(\tau )}{\tau -t}\,d\tau , \quad t\in \mathbb {R}_+, \end{aligned}$$ $$\begin{aligned} (S_2 f)(t):=\frac{1}{\pi i}\int \limits _0^\infty \left( \frac{t}{\tau }\right) ^{1/2-1/p}\frac{f(\tau )}{\tau -t}\,d\tau , \quad t\in \mathbb {R}_+, \end{aligned}$$ \(\alpha ',\beta '\) \(\alpha ',\beta '\) \(c,d\) \(c,d\) \(\mathbb {R}_+\) \(\mathbb {R}_+\) \(0\) \(0\) \(\infty \) \(\infty \) $$\begin{aligned} \limsup _{t\rightarrow s} c(t) <1, \quad \limsup _{t\rightarrow s} d(t) <1, \quad s\in \{0,\infty \}, \end{aligned}$$ $$\begin{aligned} \limsup _{t\rightarrow s} c(t) <1, \quad \limsup _{t\rightarrow s} d(t) <1, \quad s\in \{0,\infty \}, \end{aligned}$$ \((I-cU_\alpha )P_2^++(I-dU_\beta )P_2^-\) \((I-cU_\alpha )P_2^++(I-dU_\beta )P_2^-\) \(L^p(\mathbb {R}_+)\) \(L^p(\mathbb {R}_+)\)