The maximal operator in weighted variable spaces Lp(⋅)

Abstract

We study the boundedness of the maximal operator in the weighted spaces Lp(⋅)(ρ) over a bounded open set Ω in the Euclidean space ℝn or a Carleson curve Γ 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 Ap 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 ρ(t)=∏k=1mwk(|t-tk|), tk∈Γ, where wk has the property that r1p(tk)wk(r)∈Φ10, where Φ10 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 wk (similar in a sense to the Boyd indices for the Young functions defining Orlich spaces).