Recent progress in studying the boundedness of classical operators of real analysis in general Morrey-type spaces.~I

V. I. Burenkov

Eurasian mathematical journal, Tome 3 (2012) no. 3, pp. 11-32.

Résumé

The survey is aimed at providing detailed information about recent results in the problem of the boundedness in general Morrey-type spaces of various important operators of real analysis, namely of the maximal operator, fractional maximal operator, Riesz potential, singular integral operator, Hardy operator. The main focus is on the results which contain, for a certain range of the numerical parameters, necessary and sufficient conditions on the functional parameters characterizing general Morrey-type spaces, ensuring the boundedness of the aforementioned operators from one general Morrey-type space to another one. The major part of the survey is dedicated to the results obtained by the author jointly with his co-authores A. Gogatishvili, M. L. Goldman, H. V. Guliyev, V. S. Guliyev, P. Jain, R. Mustafaev, E. D. Nursultanov, R. Oinarov, A. Serbetci, T. V. Tararykova. Part I of the survey contains discussion of the definition and basic properties of the local and global general Morrey-type spaces, of embedding theorems, and of the boundedness properties of the maximal operator. Part II of the survey will contain discussion of boundedness properties of the fractional maximal operator, Riesz potential, singular integral operator, commutators of singular integral operator, Hardy operator. It will also contain discussion of interpolation theorems, of methods of proofs and of open problems.

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