The maximal operator in weighted variable spaces LP(•)

We study, the boundedness of the maximal operator in the weighted spaces L-p(. ) (rho) over a bounded open set Omega in the Euclidean space R-n or a Carleson curve Gamma in a complex plane. The weight function may belong to a certain version of a general Muckenhoupt-type condition, which is narrower than the expected Muckenhoupt condition for variable exponent, but coincides with the usual Muckenhoupt class A(p) in the case of constant p. In the case of Carleson curves there is also considered another class of weights of radial type of the form rho(t) = Pi(m)(k=1) w(k) (|t - t(k)|), t(k)= epsilon Gamma, where w(k) has the property that r/p(t(k)) w(k) (r) epsilon Phi(0)(1) is a certain Zygmund-Bari-Stechkin-type class. It is assumed that the exponent p(t) satisfies the Dini-Lipschitz condition. For such radial type weights the final statement on the boundedness is given in terms of the index numbers of the functions w(k) (similar in a sense to the Boyd indices for the Young functions defining Orlich spaces).