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Zbl 1211.42018
Persson, Lars-Erik; Samko, Natasha
Weighted Hardy and potential operators in the generalized Morrey spaces. (English)
[J] J. Math. Anal. Appl. 377, No. 2, 792-806 (2011). ISSN 0022-247X

The authors study the weighted $p\rightarrow q$-boundedness of the multi-dimensional Hardy type operators in the generalized Morrey spaces $\cal L^{p,\varphi }(\Bbb R^n, w)$ defined by an almost increasing function $\varphi (r)$ and radial type weight $w(|x|)$ from the Bary-Stechkin-type class. Sufficient conditions are obtained, in terms of some integral inequalities imposed on $\varphi $ and $w$, for such a $p\rightarrow q$-boundedness. In the case of local spaces the obtained conditions are also necessary. These results are applied to derive a similar weighted $p\rightarrow q$-boundedness of the Riesz potential operator.
[Lubomira Softova (Aversa)]

MSC 2000:: *42B25 Maximal functions
47B38 Operators on function spaces
47G40

Keywords: generalized Morrey spaces; weighted Hardy operator; weighted Hardy inequalities; Riesz potential operator; Bary-Stechkin classes; Matuszewska-Orlicz indices

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Zentralblatt MATH Berlin [Germany]