Weighted Hardy and singular operators in Morrey spaces

2008/08/18 by Samko, Natasha
#42B25 #42B35 #46E30 #47B38 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.0808.2390

Abstract

We study the weighted boundedness of the Cauchy singular integral operator S_\Gm in Morrey spaces L^p,λ(\Gm) on curves satisfying the arc-chord condition, for a class of "radial type" almost monotonic weights. The non-weighted boundedness is shown to hold on an arbitrary Carleson curve. We show that the weighted boundedness is reduced to the boundedness of weighted Hardy operators in Morrey spaces L^p,λ(0,ℓ), ℓ>0. We find conditions for weighted Hardy operators to be bounded in Morrey spaces. To cover the case of curves we also extend the boundedness of the Hardy-Littlewood maximal operator in Morrey spaces, known in the Euclidean setting, to the case of Carleson curves. Key words and phrases: Morrey space, singular operator, Hardy operator, Hardy-Littlewood maximal operator, weighted estimate.

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