Geodesic
On maximal operators associated with singular hypersurfaces
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2024), pp. 69-76 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Maximal operators associated with singular hypersurfaces in multidimensional Euclidean spaces are considered. We prove the boundedness of these operators and define a critical exponent in the space of summable functions, when hypersurfaces are given by parametric equations.
Keywords: maximal operator, averaging operator, fractional power series, critical exponent
Mots-clés : singular hypersurface. 
@article{IVM_2024_1_a4,
     author = {S. E. Usmanov},
     title = {On maximal operators associated with singular hypersurfaces},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {69--76},
     year = {2024},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2024_1_a4/}
}
TY  - JOUR
AU  - S. E. Usmanov
TI  - On maximal operators associated with singular hypersurfaces
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2024
SP  - 69
EP  - 76
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/IVM_2024_1_a4/
LA  - ru
ID  - IVM_2024_1_a4
ER  - 
%0 Journal Article
%A S. E. Usmanov
%T On maximal operators associated with singular hypersurfaces
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2024
%P 69-76
%N 1
%U http://geodesic.mathdoc.fr/item/IVM_2024_1_a4/
%G ru
%F IVM_2024_1_a4
S. E. Usmanov. On maximal operators associated with singular hypersurfaces. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2024), pp. 69-76. http://geodesic.mathdoc.fr/item/IVM_2024_1_a4/

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

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

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

[4] Sogge C.D., “Maximal operators associated to hypersurfaces with one nonvanishing principal curvature”, Fourier analysis and partial differential equations, Stud. Adv. Math., 1995, 317–323 | MR | Zbl

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

[6] Iosevich A., Sawyer E., “Maximal averages over surfaces”, Adv. Math., 132:1 (1997), 46–119 | DOI | MR | Zbl

[7] Iosevich A., Sawyer E., Seeger A., “On averaging operators associated with convex hypersurfaces of finite type”, J. Anal. Math., 79 (1999), 159–187 | DOI | MR | Zbl

[8] Ikromov I.A., Kempe M., Müller D., “Estimates for maximal functions associated to hypersurfaces in $\mathbb{R}^{3}$ and related problems of harmonic analysis”, Acta Math., 204:2 (2010), 151–271 | DOI | MR | Zbl

[9] Ikromov I.A., Usmanov S.E., “On the boundedness of maximal operators associated with hypersurfaces”, J. Math. Sci., 264:6 (2022), 715–745 | DOI | MR | Zbl

[10] Usmanov S.E., “Ob ogranichennosti maksimalnykh operatorov, svyazannykh s singulyarnymi poverkhnostyami”, Izv. vuzov. Matem., 2021, no. 6, 84–94 | Zbl

[11] Usmanov S.E., “O probleme ogranichennosti maksimalnykh operatorov”, Izv. vuzov. Matem., 2022, no. 4, 84–94 | Zbl

[12] Collins T., Greenleaf A., Pramanik M., “A multi-dimensional resolution of singularities with applications to analysis”, Amer. J. Math., 135:5 (2013), 1179–1252 | DOI | MR | Zbl

[13] Dubrovin B.A., Novikov S.P., Fomenko A.T., Sovremennaya geometriya, Nauka, M., 1979

[14] Bryuno A.D., Stepennaya geometriya v algebraicheskikh i differentsialnykh uravneniyakh, Nauka, Fizmatlit, M., 1998