Oscillatory integrals for Mittag-Leffler functions with two variables

Ikromov, Isroil A.; Ruzhansky, Michael; Safarov, Akbar R.

doi:10.5802/crmath.597

Article de recherche - Analyse harmonique

Oscillatory integrals for Mittag-Leffler functions with two variables
[Intégrales oscillatoires pour les fonctions de Mittag-Leffler à deux variables]

Ikromov, Isroil A.^1,² ; Ruzhansky, Michael ^3,⁴ ; Safarov, Akbar R.⁵

¹Institute of Mathematics of the Academy of Sciences of Uzbekistan, Olmazor district, University 46, Tashkent, Uzbekistan, Samark
²State University, Department of Mathematics, 15 University Boulevard, Samarkand, 140104, Uzbekistan
³Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, Ghent, Belgium
⁴School of Mathematical Sciences, Queen Mary University of London, United Kingdom
⁵Uzbek-Finnish Pedagogical Institute, Spitamenshox 166, Samarkand, Uzbekistan

Comptes Rendus. Mathématique, Tome 362 (2024) no. G7, pp. 789-798

Abstract (VO)
Résumé

In this paper we consider the problem of estimation of oscillatory integrals with Mittag-Leffler functions in two variables. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study oscillatory type integrals.

Dans cet article, nous considérons le problème de l’estimation des intégrales oscillatoires avec les fonctions de Mittag-Leffler à deux variables. La généralisation est que l’on remplace la fonction exponentielle par la fonction de type Mittag-Leffler, pour étudier les intégrales de type oscillatoire.

Reçu le : 2022-12-16
Révisé le : 2023-12-01
Accepté le : 2023-12-02
Publié le : 2024-09-17

DOI : 10.5802/crmath.597

Classification : 35D10, 42B20, 26D10

Affiliations des auteurs :

Ikromov, Isroil A. ^{1
,

2} ; Ruzhansky, Michael ^{3
,

4} ; Safarov, Akbar R. ⁵

¹ Institute of Mathematics of the Academy of Sciences of Uzbekistan, Olmazor district, University 46, Tashkent, Uzbekistan, Samark
² State University, Department of Mathematics, 15 University Boulevard, Samarkand, 140104, Uzbekistan
³ Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, Ghent, Belgium
⁴ School of Mathematical Sciences, Queen Mary University of London, United Kingdom
⁵ Uzbek-Finnish Pedagogical Institute, Spitamenshox 166, Samarkand, Uzbekistan

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Ikromov, Isroil A. and Ruzhansky, Michael and Safarov, Akbar R.},
     title = {Oscillatory integrals for {Mittag-Leffler} functions with two variables},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {789--798},
     year = {2024},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     number = {G7},
     doi = {10.5802/crmath.597},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/crmath.597/}
}

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Ikromov, Isroil A.; Ruzhansky, Michael; Safarov, Akbar R. Oscillatory integrals for Mittag-Leffler functions with two variables. Comptes Rendus. Mathématique, Tome 362 (2024) no. G7, pp. 789-798. doi: 10.5802/crmath.597

Bibliographie
Cité par

[1] Agarwal, Ratan Prakash À propos d’une note de M. Pierre Humbert, C. R. Acad. Sci. Paris, Volume 236 (1953), pp. 2031-2032 | MR | Zbl

[2] Arnol’d, V. I.; Gusejn-Zade, S. M.; Varchenko, A. N. Singularities of Differentiable Maps. Volume I: The classification of critical points, caustics and wave fronts, Monographs in Mathematics, 82, Birkhäuser, Boston-Basel-Stuttgart, 1985 | MR | Zbl | DOI

[3] Arkhipov, G. I.; Karatsuba, A. A.; Chubarikov, V. N. Theory of multiple trigonometric sums, Nauka, Moska, 1987 | Zbl

[4] Dzherbashyan, M. M. On Abelian summation of the eneralized integral transform, Akad. Nauk Armjan. SSR Izvestija, fiz-mat. estest. techn. nauki, Volume 7 (1954), pp. 1-26

[5] Dzherbashyan, M. M. On integral representation of functions continuous on given rays (generalization of the Fourier integrals), Izv. Akad. Nauk SSSR, Ser. Mat., Volume 18 (1954), pp. 427-448

[6] Dzherbashyan, M. M. On the asymtotic expansion of a function of Mittag-Leffler type, Akad. Nauk Armjan. SSR Doklady, Volume 19 (1954), pp. 65-72

[7] Gorenflo, R.; Kilbas, A. A.; Mainardi, F.; Rogosin, S. V. Mittag–Leffler functions, related topics and applications, Springer Monographs in Mathematics, Springer, 2014 | MR | DOI | Zbl

[8] Greenblat, M. Oscillatory integral decay, sublevel set growth and the Newton polyhedron, Math. Ann., Volume 346 (2010) no. 4, pp. 857-890 | MR | DOI | Zbl

[9] Green, J. Uniform oscillatory integral estimates for convex phases via sublevel set estimates (2021) (2111.05395v1)

[10] Humbert, P.; Agarwal, R. P. Sur la fonction de Mittag-Leffler et quelques-unes de ses généralisations, Bull. Sci. Math., Volume 77 (1953), pp. 180-185 | MR | Zbl

[11] Humbert, P. Quelques résultats relatifs à la fonction de Mittag-Leffler, C. R. Acad. Sci. Paris, Volume 236 (1953), pp. 1467-1468 | MR | Zbl

[12] Ikromov, I. A.; Kempe, M.; Müller, D. Estimates for maximal functions associated with hypersurfaces in $ℝ^{3}$ and related problems of harmonic analysis, Acta Math., Volume 204 (2010) no. 2, pp. 151-271 | MR | DOI | Zbl

[13] Ikromov, I. A. Invariant estimates of two-dimensional trigonometric integrals, Math. USSR, Sb., Volume 67 (1990) no. 2, pp. 473-488 | DOI | Zbl

[14] Ikromov, Isroil A.; Müller, Detlef On adapted coordinate systems, Trans. Am. Math. Soc., Volume 363 (2011) no. 6, pp. 2821-2848 | MR | DOI | Zbl

[15] Ikromov, I. A.; Müller, D. Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra, Annals of Mathematics Studies, 194, Princeton University Press, 2016 | MR | DOI | Zbl

[16] Ikromov, Isroil A.; Safarov, Akbar R.; Absalamov, Akmal T. On the convergence exponent of the special integral of the Tarry problem for a quadratic polynomial, Zh. Sib. Fed. Univ. Mat. Fiz., Volume 16 (2023) no. 4, pp. 488-497 | MR

[17] Karpushkin, V. N. Uniform estimates of oscillating integrals in $ℝ^{2}$ , Dokl. Akad. Nauk SSSR, Volume 254 (1980) no. 1, pp. 28-31 | MR

[18] Mittag-Leffler, G. Sur l’intégrale de Laplace–Abel, C. R. Acad. Sci. Paris, Volume 135 (1902), pp. 937-939 | Zbl

[19] Mittag-Leffler, G. Sur la nouvelle fonction $E_{α} (x)$ , C. R. Acad. Sci. Paris, Volume 137 (1903), pp. 554-558 | Zbl

[20] Mittag-Leffler, G. Une généralization de l’intégrale de Laplace–Abel, C. R. Acad. Sci. Paris, Volume 136 (1903), pp. 537-539 | Zbl

[21] Mittag-Leffler, G. Sopra la funzione $E_{α} (x)$ , Rom. Acc. L. Rend. (5), Volume 13 (1904) no. 1, pp. 3-5 | Zbl

[22] Podlubny, I. Fractional Differensial Equations, Mathematics in Science and Engineering, 198, Academic Press Inc., 1999 | MR | Zbl

[23] Phong, D. H.; Stein, E. M. The Newton polyhedron and oscillatory integral operator, Acta Math., Volume 179 (1997) no. 1, pp. 105-152 | DOI | Zbl | MR

[24] Ruzhansky, M.; Safarov, A. R.; Khasanov, G. A. Uniform estimates for oscillatory integrals with homogeneous polynomial phases of degree 4, Anal. Math. Phys., Volume 12 (2022) no. 6, 130 | DOI | Zbl | MR

[25] Ruzhansky, M.; Torebek, B. T. Multidimensional van der Corput-type estimates involving Mittag-Leffler functions, Fract. Calc. Appl. Anal., Volume 23 (2021) no. 6, pp. 1663-1677 | DOI | Zbl | MR

[26] Ruzhansky, Michael; Torebek, Berikbol T. Van der Corput lemmas for Mittag-Leffler functions. II. $α -$ directions, Bull. Sci. Math., Volume 171 (2021) no. 3, 103016 | MR | DOI | Zbl

[27] Ruzhansky, M. Pointwise van der Corput Lemma for Functions of Several Variables, Funct. Anal. Appl., Volume 43 (2009) no. 1, pp. 75-77 | DOI | Zbl

[28] Ruzhansky, M. Multidimensional decay in the van der Corput Lemma, Stud. Math., Volume 208 (2012) no. 1, pp. 1-9 | DOI | Zbl | MR

[29] Safarov, A. R. On invariant estimates for oscillatory integrals with polynomial phase, J. Sib. Fed. Univ., Math. Phys., Volume 9 (2016) no. 1, pp. 102-107 | DOI | Zbl

[30] Safarov, A. R. Invariant estimates of two-dimensional oscillatory integrals, Math. Notes, Volume 104 (2018) no. 2, pp. 293-302 | DOI | Zbl | MR

[31] Safarov, A. R. On a problem of restriction of Fourier transform on a hypersurface, Russ. Math., Volume 63 (2019) no. 4, pp. 57-63 | DOI | Zbl

[32] Safarov, A. R. On the $L^{p}$ -bound for trigonometric integrals, Anal. Math., Volume 45 (2019), pp. 153-176 | DOI | Zbl | MR

[33] Safarov, A. R. Estimates for Mittag-Leffler Functions with Smooth Phase Depending on Two Variables, J. Sib. Fed. Univ., Math. Phys., Volume 15 (2022) no. 4, pp. 459-466 | Zbl

[34] Varchenko, A. N. Newton polyhedra and estimation of oscillating integrals, Funct. Anal. Appl., Volume 10 (1976), pp. 175-196 | DOI | Zbl

[35] van der Corput, J. G. Zur Methode der stationären Phase. I. Einfache Integrale, Compos. Math., Volume 1 (1934), pp. 15-38 | Zbl

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