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

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