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Zbl 1155.42005
Samko, Natasha
Weighted Hardy and singular operators in Morrey spaces. (English)
[J] J. Math. Anal. Appl. 350, No. 1, 56-72 (2009). ISSN 0022-247X

In the interesting paper under review, the author studies the weighted boundedness of the Cauchy singular integral operator $S_{\Gamma}$ in the framework of the Morrey spaces $\Cal L^{p,\lambda}(\Gamma)$ on curves satisfying the arc-chord condition, for a class of ``radial type'' almost monotonic weights. The non-weighted boundedness is shown to hold over an arbitrary Carleson curve, while the weighted boundedness is reduced to the boundedness of weighted Hardy operators in Morrey spaces $\Cal L^{p,\lambda}(0,\ell), \ell >0.$ Conditions are found for the weighted Hardy operators in order to be bounded in Morrey spaces. To cover the case of curves, the author extends the boundedness of the Hardy-Littlewood maximal operator in Morrey spaces, known in the Euclidean setting, to the case of Carleson curves.
[Lubomira Softova (Bari)]

MSC 2000:: *42B20 Singular integrals, several variables
42B25 Maximal functions

Keywords: Morrey space; Singular operator; Hardy operator; Hardy-Littlewood maximal operator; Weighted estimate

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Zentralblatt MATH Berlin [Germany]