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 CMFD, 2018, Volume 64, Issue 4, Pages 650–681 (Mi cmfd365)

On boundedness of maximal operators associated with hypersurfaces

I. A. Ikromov, S. E. Usmanov

Samarkand State University, Samarkand, Uzbekistan

Abstract: In this paper, we obtain the criterion of boundedness of maximal operators associated with smooth hypersurfaces. Also we compute the exact value of the boundedness index of such operators associated with arbitrary convex analytic hypersurfaces in the case where the height of a hypersurface in the sense of A. N. Varchenko is greater than 2. Moreover, we obtain the exact value of the boundedness index for degenerated smooth hypersurfaces, i.e., for hypersurfaces satisfying conditions of the classical Hartman–Nirenberg theorem. The obtained results justify the Stein–Iosevich–Sawyer hypothesis for arbitrary convex analytic hypersurfaces as well as for smooth degenerated hypersurfaces. Also we discuss some related problems of the theory of oscillatory integrals.

 Funding Agency Grant Number Commitee for coordination science and technologies under Cabinet Ministers of the Republic of Uzbekistan ÎÒ-Ô-4-69

DOI: https://doi.org/10.22363/2413-3639-2018-64-4-650-681

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UDC: 517.982.42

Citation: I. A. Ikromov, S. E. Usmanov, “On boundedness of maximal operators associated with hypersurfaces”, Contemporary problems in mathematics and physics, CMFD, 64, no. 4, Peoples' Friendship University of Russia, M., 2018, 650–681

Citation in format AMSBIB
\Bibitem{IkrUsm18} \by I.~A.~Ikromov, S.~E.~Usmanov \paper On boundedness of maximal operators associated with hypersurfaces \inbook Contemporary problems in mathematics and physics \serial CMFD \yr 2018 \vol 64 \issue 4 \pages 650--681 \publ Peoples' Friendship University of Russia \publaddr M. \mathnet{http://mi.mathnet.ru/cmfd365} \crossref{https://doi.org/10.22363/2413-3639-2018-64-4-650-681}