Remark on the Boundedness of the Cauchy Singular Integral Operator on Variable Lebesgue Spaces with Radial Oscillating Weights

TL;DR

This paper investigates the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with radial oscillating weights, providing a partial converse to recent sufficient conditions established in prior work.

Contribution

It offers a partial converse to the recent sufficient condition for boundedness, deepening understanding of the operator's behavior on these weighted spaces.

Findings

Established a partial converse to the boundedness condition.

Connected Matuszewska-Orlicz indices with operator boundedness.

Extended the theoretical framework for variable Lebesgue spaces with oscillating weights.

Abstract

Recently V. Kokilashvili, N. Samko, and S. Samko have proved a sufficient condition for the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with radial oscillating weights over Carleson curves. This condition is formulated in terms of Matuszewska-Orlicz indices of weights. We prove a partial converse of their result.