Annals of Functional Analysis

Nonlinear harmonic analysis of integral operators in weighted grand Lebesgue spaces and applications

Alberto Fiorenza and Vakhtang Kokilashvili

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In this article, we give a boundedness criterion for Cauchy singular integral operators in generalized weighted grand Lebesgue spaces. We establish a necessary and sufficient condition for the couple of weights and curves ensuring boundedness of integral operators generated by the Cauchy singular integral defined on a rectifiable curve. We characterize both weak and strong type weighted inequalities. Similar problems for Calderón–Zygmund singular integrals defined on measured quasimetric space and for maximal functions defined on curves are treated. Finally, as an application, we establish existence and uniqueness, and we exhibit the explicit solution to a boundary value problem for analytic functions in the class of Cauchy-type integrals with densities in weighted grand Lebesgue spaces.

Article information

Ann. Funct. Anal., Volume 9, Number 3 (2018), 413-425.

Received: 6 June 2017
Accepted: 5 September 2017
First available in Project Euclid: 6 February 2018

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Zentralblatt MATH identifier

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 42B25: Maximal functions, Littlewood-Paley theory 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Cauchy singular integral operator Carleson curve Muckenhoupt $A_{p}$ class Calderón–Zygmund singular integrals Riemann boundary value problem


Fiorenza, Alberto; Kokilashvili, Vakhtang. Nonlinear harmonic analysis of integral operators in weighted grand Lebesgue spaces and applications. Ann. Funct. Anal. 9 (2018), no. 3, 413--425. doi:10.1215/20088752-2017-0056.

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