Abstract
We study the boundedness of the maximal operator in the weighted spacesLp(⋅)(ρ)over a bounded open setΩin the Euclidean spaceℝnor a Carleson curveΓin 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 classApin the case of constantp. In the case of Carleson curves there is also considered another class of weights of radial type of the formρ(t)=∏k=1mwk(|t-tk|),tk∈Γ, wherewkhas the property thatr1p(tk)wk(r)∈Φ10, whereΦ10is a certain Zygmund-Bari-Stechkin-type class. It is assumed that the exponentp(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 functionswk(similar in a sense to the Boyd indices for the Young functions defining Orlich spaces).
Highlights
Within the frameworks of variable exponent spaces Lp(·)(Ω), the boundedness of maximal operators was proved in L
Because of applications to weighted boundedness of singular integral operator along Carleson curves, we prove similar results for the maximal operator along Carleson curves
This extension from the Euclidean space to the case of Carleson curves required an essential modification of certain means used in [10]
Summary
Within the frameworks of variable exponent spaces Lp(·)(Ω), the boundedness of maximal operators was proved in L. A certain subclass of general weights was considered in [10], where for the case of bounded domains Ω in the Euclidean space, the boundedness of the maximal operator in the spaces Lp(·)(Ω, ρ) was proved. This subclass may be characterized as a class of radial type weights which satisfy the Zygmund-Bari-Stechkin condition. Because of applications to weighted boundedness of singular integral operator along Carleson curves, we prove similar results for the maximal operator along Carleson curves This extension from the Euclidean space to the case of Carleson curves required an essential modification of certain means used in [10].
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