Geodesic
On the sharp estimates for maximal operators
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 16 (2023) no. 3, pp. 348-356.

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The paper deals with boundedness problem for the maximal operators associated with hypersurfaces in the space of square integrable functions. A necessary condition for boundedness is given in the case of one nonvanishing principal curvature. A criterion for the boundedness is obtained for a particular class of convex hypersurfaces.
Keywords: maximal operator, boundedness.
Mots-clés : Fourier transform, hypersurface 
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Azamat M. Barakayev. On the sharp estimates for maximal operators. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 16 (2023) no. 3, pp. 348-356. http://geodesic.mathdoc.fr/item/JSFU_2023_16_3_a6/

[1] E.M.Stein, “Maximal functions.I.Spherical means”, Proc.Nat. Acad. Sci.U.S.A., 73:7 (1976), 2174–2175 | DOI | MR | Zbl

[2] J.Bourgain, “Averages in the plane convex cuves and maximal operators”, J. Anal. Math., 47:7 (1986), 69–85 | DOI | MR | Zbl

[3] A.Iosevich, E.Sawyer, “Maximal Averages over surfaces”, Adv. in Math., 132 (1997), 119–187 | DOI | MR

[4] A.Iosevich, E.Sawyer, “Oscillatory integrals and maximal averages over homogeneous surfaces.”, Duke Math., 82:1 (1996), 103–131 | DOI | MR

[5] I.A.Ikromov, M.Kempe, D.Müller, “Estimates for maximal functions assosiated to hypersurfaces in $R^3$ and related problems of harmonic analysis”, Acta Math., 204 (2010), 151–271 | DOI | MR | Zbl

[6] A.N.Varchenko, “Newton's polytopes and estimates of oscillating integrals”, Functional analysis and its applications, 10:3 (1976), 13–38 | MR

[7] C.D.Sogge, “Maximal operators associated to hypersurfaces with one nonvanishing principal curvature”, Fourier Analysis and Partial Differential Equations, Stud. Adv. Math., 1995, 317–323 | MR | Zbl

[8] I.A.Ikromov, S.E.Usmanov, “On the Boundedness of Maximal Operations Associated with Hypersurfaces”, Modern Mathematics Fundamental Directions, 64:4 (2018), 650–681 (in Russian) | DOI | MR

[9] H.Schulz, “Convex hypersurfaces of finite type and the asymptotics of their Fourier transforms”, Indiana Univ. Math. J., 40:4 (1999), 1267–1275 | DOI | MR

[10] I.A.Ikromov, D.Müller, Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra, AM, 194, Princeton University Press, 2016 | DOI | MR | Zbl

[11] A.Greenleaf, “Principal curvature and harmonic analysis”, Indiana Univ. Math. J., 30:4 (1981), 519–537 | DOI | MR | Zbl

[12] A.R.Safarov, “On a problem of restriction of Fourier transform on a hypersurface”, Russian Mathematics, 63:4 (2019), 57–63 (in Russian) | DOI | MR | Zbl

[13] S.Buschenhenke, I.A.Ikromov, D.Muller, Estimates for Maximal functions associated to hypersurfaces in $R^3$ with height $h 2$, 2022, arXiv: 2209.07352 [math.CA] | DOI | Zbl

[14] I.A.Ikromov, A.M.Barakayev, On boundedness of maximal operators, Preprint Izvestiay Vuzov (in Russian)

[15] C.D.Sogge, E.M.Stein, “Averages of functions over hypersurfaces in $R^{n}$”, Invent. Math., 82:3 (1985), 543–556 | DOI | MR | Zbl

[16] I.A.Ikromov, M.Kempe, D.Muller, “Damped oscillatory integrals and boundedness of maximal operators associated to mixed homogeneous hypersurfaces”, Duke Math. J., 126:3 (2005), 471–490 | DOI | MR | Zbl

[17] A.Nagel, A.Seeger, S.Wainger, “Averages over convex hypersurfaces”, Amer. J. Math., 115:4 (1993), 903–927 | DOI | MR | Zbl