Oscillatory integrals for Mittag-Leffler functions with two variables

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

doi:10.5802/crmath.597

Research article - Harmonic analysis

Oscillatory integrals for Mittag-Leffler functions with two variables

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

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 789-798.

Abstract
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.

Received: 2022-12-16
Revised: 2023-12-01
Accepted: 2023-12-02
Published online: 2024-09-17

DOI: 10.5802/crmath.597

Classification: 35D10, 42B20, 26D10

Author's affiliations:

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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

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Isroil A. Ikromov; Michael Ruzhansky; Akbar R. Safarov. Oscillatory integrals for Mittag-Leffler functions with two variables. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 789-798. doi : 10.5802/crmath.597. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.597/

References
Cited by

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

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

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

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

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

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

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

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

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

[12] I. A. Ikromov; M. Kempe; D. Müller 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 | DOI | Zbl

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

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

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

[16] Isroil A. Ikromov; Akbar R. Safarov; Akmal T. Absalamov 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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