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arXiv · 0808.2390

Weighted Hardy and singular operators in Morrey spaces

Natasha Samko
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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 $L^{p,λ}(0,\ell), \ell>0$. 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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Natasha Samko. 2008-08-18. Weighted Hardy and singular operators in Morrey spaces. https://arxiv.org/abs/0808.2390

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