Corrections to the paper "The boundedness of certain sublinear operator in the weighted variable Lebesgue spaces"

TL;DR

This paper corrects a previously published proof regarding the boundedness of certain sublinear operators in weighted variable Lebesgue spaces, providing accurate conditions and a revised theorem to ensure validity.

Contribution

The authors identify and correct a significant error in their earlier proof, offering a revised theorem with accurate conditions for boundedness in weighted variable Lebesgue spaces.

Findings

Corrected the proof of boundedness for sublinear operators

Provided accurate conditions for two-weight inequalities

Revised Theorem 5 with a valid argument

Abstract

In this paper author was proved the boundedness of the multidimensional Hardy type operator in weighted Lebesgue spaces with a variable exponent. As an application we prove the boundedness of certain sublinear operators on the weighted variable Lebesgue space. Note that the proof of multidimensional Hardy type operator in weighted Lebesgue spaces with a variable exponent don't contained any mistakes. But at the proving of the boundedness of certain sublinear operators on the weighted variable Lebesgue space Georgian colleagues discovered a small but significant error in this paper. This result is assigned as Theorem 5 in noted paper. In other words, sufficient conditions for general weights ensuring the validity of the two-weight strong type inequalities for some sublinear operator was found. In this theorem the inequality (9) isn't true. In this note we give the details of the correct argument. We presume that the reader is familiar with the contents and notation of our original paper. At the heart of our correction is the following Theorem which replaces Theorem 5.