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 ϕt,γ(τ ) = |(τ − t) γ |, where γ is a complex number, over arbitrary Carleson curves. If the curve has different spirality indices at the point t and γ is not real, then ϕt,γ is an oscillating weight lying beyond the class of radial oscillating weights considered recently by V. Kokilashvili, N. Samko, and S. Samko.

Potential operators on carleson curves in morrey spaces

“…Maximal operators and potential operators in various spaces de…ned on Carleson curves has been widely studied by many authors (see, for example [2,3,6,7,8,9,10,12]). N. Samko [12] studied the boundedness of the maximal operator M de…ned on quasimetric measure spaces, in particular on Carleson curves in Morrey spaces L p; ( ): Theorem A.…”