On a weighted singular integral operator with shifts and slowly oscillating data

DOI10.1007/S11785-015-0452-0zbMath1352.45005arXiv1501.03744OpenAlexW2094457575MaRDI QIDQ317794

Amarino B. Lebre, Alexei Yu. Karlovich, Yuri I. Karlovich

Publication date: 4 October 2016

Published in: Complex Analysis and Operator Theory (Search for Journal in Brave)

Abstract: Let $$ be orientation-preserving diffeomorphism (shifts) of $mathbb{R}_{+}=(0,infty)$ onto itself with the only fixed points $0$ and $infty$ and $$ be the isometric shift operators on $L^p(mathbb{R}_{+})$ given by $U_{a}lphaf=(alph{a}^{\prime }{)}^{1/p}(fcircalpha)$, $$, and $P_2^{p}m=(Ipm{S}_2)/2$ where [ (S_2 f)(t):=frac{1}{pi i}intlimits_0^infty left(frac{t}{ au} ight)^{1/2-1/p}frac{f( au)}{ au-t},d au, quad tinmathbb{R}_+, ] is the weighted Cauchy singular integral operator. We prove that if $$ and $c,d$ are continuous on $mathbb{R}_{+}$ and slowly oscillating at $0$ and $infty$, and [ limsup_{t o s}|c(t)|<1, quad limsup_{t o s}|d(t)|<1, quad sin{0,infty}, ] then the operator $$ is Fredholm on $L^p(mathbb{R}_{+})$ and its index is equal to zero. Moreover, its regularizers are described.

Full work available at URL: https://arxiv.org/abs/1501.03744

zbMATH Keywords

indexFredholmnessorientation-preserving shiftslowly oscillating functionweighted Cauchy singular integral operator

Mathematics Subject Classification ID

Toeplitz operators, Hankel operators, Wiener-Hopf operators (47B35) (Semi-) Fredholm operators; index theories (47A53) Integral operators (47G10) Integral equations with kernels of Cauchy type (45E05) Pseudodifferential operators (47G30)

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• The index of weighted singular integral operators with shifts and slowly oscillating data
• Non-commutative Gelfand theories. A tool-kit for operator theorists and numerical analysts
• Sufficient conditions for Fredholmness of singular integral operators with shifts and slowly oscillating data
• Necessary conditions for Fredholmness of singular integral operators with shifts and slowly oscillating data
• On algebras generated by convolutions and discontinuous functions
• Spectral theory of approximation methods for convolution equations
• Limit operators and their applications in operator theory
• Linear functional equations. Operator approach. Transl. from the Russian by V. Muzafarov a. A. Iacob
• On regularization of Mellin PDO's with slowly oscillating symbols of limited smoothness
• Fredholmness and index of simplest singular integral operators with two slowly oscillating shifts
• AN ALGEBRA OF PSEUDODIFFERENTIAL OPERATORS WITH SLOWLY OSCILLATING SYMBOLS

Related Items (9)

The Haseman boundary value problem with quasicontinuous coefficients and shifts ⋮ The generalized Cauchy index of some semi-almost periodic functions ⋮ Closed-form solutions for several classes of singular integral equations with convolution and Cauchy operator ⋮ On Singular Integral Operators with Shifts ⋮ The index of weighted singular integral operators with shifts and slowly oscillating data ⋮ Singular Integral Equations of Convolution Type with Reflection and Translation Shifts ⋮ Solving convolution singular integral equations with reflection and translation shifts utilizing Riemann-Hilbert approach ⋮ Mellin pseudodifferential operators and singular integral operators with complex conjugation ⋮ Semi-Fredholmness of Weighted Singular Integral Operators with Shifts and Slowly Oscillating Data

This page was built for publication: On a weighted singular integral operator with shifts and slowly oscillating data