Maximal operators on variable Lebesgue spaces with weights related to oscillations of Carleson curves

TL;DR

This paper establishes conditions for the boundedness of the maximal operator on variable Lebesgue spaces with oscillating weights related to Carleson curves, extending previous results to more complex weights and curves.

Contribution

It introduces new sufficient conditions for boundedness involving weights with oscillations tied to Carleson curves and complex exponents, broadening the scope of prior work.

Findings

Boundedness conditions depend on spirality indices of curves.

Oscillating weights with complex exponents are analyzed.

Results extend to weights beyond radial oscillating classes.

Abstract

We prove sufficient conditions for the boundedness of the maximal operator on variable Lebesgue spaces with weights $\varphi_{t,\gamma}(\tau)=|(\tau-t)^\gamma|$, where $\gamma$ is a complex number, over arbitrary Carleson curves. If the curve has different spirality indices at the point $t$ and $\gamma$ is not real, then $\varphi_{t,\gamma}$ is an oscillating weight lying beyond the class of radial oscillating weights considered recently by V. Kokilashvili, N. Samko, and S. Samko.