## Annals of Functional Analysis

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

#### Abstract

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

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

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

https://projecteuclid.org/euclid.afa/1517886227

Digital Object Identifier
doi:10.1215/20088752-2017-0056

Mathematical Reviews number (MathSciNet)
MR3835228

Zentralblatt MATH identifier
06946365

#### Citation

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. https://projecteuclid.org/euclid.afa/1517886227

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