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Weighted Hardy Spaces Over the Unit Ball: The Freely Noncommutative and Commutative Settings Read chapter Read first chapter

2024 | OriginalPaper | Chapter

Maximal Noncompactness of Singular Integral Operators on \(L^2\) Spaces with Some Khvedelidze Weights

Authors : Oleksiy Karlovych, Alina Shalukhina

Published in: Operator and Matrix Theory, Function Spaces, and Applications

Publisher: Springer Nature Switzerland

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Abstract

Let \(\Gamma \) be a contour in the complex plane consisting of a finite number of circular arcs joining the endpoints \(-1\) and 1, possibly including the segment \([-1,1]\). We consider the singular integral operator \(A=aI+bS_\Gamma \) with constant coefficients \(a,b\in \mathbb {C}\), where \(S_\Gamma \) is the Cauchy singular integral operator over \(\Gamma \). We provide a detailed proof of the maximal noncompactness of the operator A on \(L^2\) spaces with the Khvedelidze weights \(\varrho (t)=|t-1|{ }^\beta |t+1|{ }^{-\beta }\) satisfying \(-1<\beta <1\). This result was announced by Naum Krupnik in 2010, but its proof has never been published.

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previous chapter Mellin Pseudodifferential Operators and Singular Integral Operators with Complex Conjugation
next chapter On the Dual Representation of the Congruence Kernels and the Related Delsarte Type Transmutations of Multidimensional Differential Operators
Literature
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R. Avendaño, N. Krupnik, On the connection between the norm and the essential norm of the singular operator. Doklady Rasshirennyh Zasedaniy Seminara Instituta Prikladnoy Matematiki im. I. N. Vekua, 3:5–8, 1988 (in Russian)
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M. Bertola, A. Katsevich, A. Tovbis, On the spectral properties of the Hilbert transform operator on multi-intervals. J. Spectr. Theory 12(2), 339–379 (2022)MathSciNetCrossRef
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O. Karlovych, E. Shargorodsky, Discrete Riesz transforms on rearrangement-invariant Banach sequence spaces and maximally noncompact operators. Pure Appl. Functional Anal. (2024), to appear
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N. Krupnik, Banach Algebras with Symbol and Singular Integral Operators, volume 26 of Operator Theory: Advances and Applications (Birkhäuser Verlag, Basel, 1987)CrossRef
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N. Krupnik, Survey on the best constants in the theory of one-dimensional singular integral operators, in Topics in Operator Theory. Volume 1. Operators, Matrices and Analytic Functions, volume 202 of Operator Theory: Advances and Applications (Birkhäuser Verlag, Basel, 2010), pp. 365–393
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Metadata
Title
Maximal Noncompactness of Singular Integral Operators on Spaces with Some Khvedelidze Weights
Authors
Oleksiy Karlovych
Alina Shalukhina
Copyright Year
2024
Book
Operator and Matrix Theory, Function Spaces, and Applications

Print ISBN: 978-3-031-50612-3

Electronic ISBN: 978-3-031-50613-0

Copyright Year: 2024

DOI
https://doi.org/10.1007/978-3-031-50613-0_12

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