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

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.