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

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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 ∞.

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Authors and Affiliations

  1. Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829-516, Caparica, Portugal

    Alexei Yu. Karlovich

  2. Facultad de Ciencias, Universidad Autónoma del Estado de Morelos, Av. Universidad 1001, Col. Chamilpa, C.P. 62209, Cuernavaca, Morelos, México

    Yuri I. Karlovich

  3. Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais, 1049-001, Lisboa, Portugal

    Amarino B. Lebre

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  1. Alexei Yu. Karlovich
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  2. Yuri I. Karlovich
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  3. Amarino B. Lebre
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Correspondence to Alexei Yu. Karlovich.

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This work is partially supported by “Centro de Análise Funcional e Aplicações” at Instituto Superior Técnico (Lisboa, Portugal), which is financed by FCT (Portugal). The second author is also supported by the SEP-CONACYT Project No. 25564 (México) and by PROMEP (México) via “Proyecto de Redes”.

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Karlovich, A.Y., Karlovich, Y.I. & Lebre, A.B. Sufficient Conditions for Fredholmness of Singular Integral Operators with Shifts and Slowly Oscillating Data. Integr. Equ. Oper. Theory 70, 451–483 (2011). https://doi.org/10.1007/s00020-010-1861-0

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  • DOI: https://doi.org/10.1007/s00020-010-1861-0

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