V. I. Burenkov, V. S. Guliyev, A. Serbetci, T. V. Tararykova, “Necessary and sufficient conditions for the boundedness of genuine singular integral operators in local Morrey-type spaces”, Eurasian Math. J., 2010, Volume 1, Number 1,Pages <nobr>32

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Necessary and sufficient conditions for the boundedness of genuine singular integral operators in local Morrey-type spaces

V. I. Burenkov^a, V. S. Guliyev^b, A. Serbetci^c, T. V. Tararykova^a

^a Faculty of Mathematics and Information Technology, L. N. Gumilyov Eurasian National University, Astana, Kazakhstan
^b Institute of Mathematics and Mechanics, Academy of Sciences of Azerbaijan, Baku, Azerbaijan
^c Ankara University, Department of Mathematics, Tandogan-Ankara, Turkey

Abstract: The problem of the boundedness of a Calderon-Zygmund singular integral operator $T$ in local Morrey-type spaces is reduced to the boundedness of the Hardy operator in weighted $L_p$-spaces on the cone of non-negative non-increasing functions. This allows obtaining sufficient conditions for the boundedness of $T$ in local Morrey-type spaces for all admissible values of the parameters. Moreover, for a certain range of the parameters, for a genuine Calderon-Zygmund singular integral operator these sufficient conditions coincide with the necessary ones.

Keywords and phrases: singular integral operator, maximal operator, local Morrey-type spaces, Hardy operator on the cone of monotonic functions, weak Morrey-type spaces, weighted estimates.

MSC: 42B20, 42B25, 42B35

Received: 01.10.2009

Language: English