Geodesic
On the boundedness problem of maximal operators
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 4 (2022), pp. 84-94.

Voir la notice de l'article provenant de la source Math-Net.Ru

It is considered maximal operators associated with singular surfaces in the space $\mathbb{R}^{3}.$ It is proved theorem on the boundedness of these operators in $L^{p}$, when a singular surface is given by parametric equations.
Keywords: maximal operator, averaging operator, fractional power series, indicator of boundedness, singular surface. 
@article{IVM_2022_4_a6,
     author = {S. E. Usmanov},
     title = {On the boundedness problem of maximal operators},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {84--94},
     publisher = {mathdoc},
     number = {4},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2022_4_a6/}
}
TY  - JOUR
AU  - S. E. Usmanov
TI  - On the boundedness problem of maximal operators
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2022
SP  - 84
EP  - 94
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2022_4_a6/
LA  - ru
ID  - IVM_2022_4_a6
ER  - 
%0 Journal Article
%A S. E. Usmanov
%T On the boundedness problem of maximal operators
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2022
%P 84-94
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2022_4_a6/
%G ru
%F IVM_2022_4_a6
S. E. Usmanov. On the boundedness problem of maximal operators. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 4 (2022), pp. 84-94. http://geodesic.mathdoc.fr/item/IVM_2022_4_a6/

[1] Stein E. M., “Maximal functions: Spherical means”, Proc. Natl. 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 Anal. and Partial Diff. Equat., Stud. Adv. Math., CRC, Boca Raton, FL, 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., “Oscillatory integrals and maximal averages over homogeneous surfaces”, Duke Math. J., 82:1 (1996), 103–141 | DOI | MR | Zbl

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

[8] 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

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

[10] 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

[11] Ikromov I. A., “Dempfirovannye ostsillyatornye integraly i maksimalnye operatory”, Matem. zametki, 78:6 (2005), 833–852 | Zbl

[12] Greenblatt M., “$L^p$ boundedness of maximal averages over hypersurfaces in $\mathbb{R}^3$”, Trans. Amer. Math. Soc., 365:4 (2013), 1875–1900 | DOI | MR | Zbl

[13] Ikromov I. A., Usmanov S. E., “Ob ogranichennosti maksimalnykh operatorov, svyazannykh s giperpoverkhnostyami.”, Sovremen. matem. Fundament. napravleniya, 64, no. 4, 2018, 650–681 | MR

[14] Usmanov S. E., “Ogranichennost maksimalnykh operatorov, svyazannykh s singulyarnymi poverkhnostyami”, DAN Resp. Uzbekistan, 4:4 (2017), 40–45

[15] Usmanov S. E., “On properties of some singular surfaces”, Scientific J. Samarkand Univ., 2020:3 (2020), 64–70

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

[17] 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

[18] Greenblatt M., “A direct resolution of singularities for functions of two variables with applications to analysis”, J. Anal. Math., 92 (2004), 233–257 | DOI | MR | Zbl