Weierstrass polynomials in estimates of oscillatory integrals

I. A. Ikromov ; A. S. Sadullaev

Contemporary Mathematics. Fundamental Directions, Science — Technology — Education — Mathematics — Medicine, Tome 67 (2021) no. 4, pp. 668-692.

Résumé

In this paper, estimates are obtained for the Fourier transform of smooth charges (measures) concentrated on some nonconvex hypersurfaces. The summability of the maximal Randall function is proved for a wide class of nonconvex hypersurfaces. In addition, in the three-dimensional case, estimates are obtained depending on the Varchenko height. The accuracy of the obtained estimates is proved. The proof of the estimate for oscillatory integrals is based on the Weierstrass preparatory theorem.

Bibliographie

[1] Arnold V. I., Varchenko A. N., Gusein-zade S. M., Osobennosti differntsiruemykh otobrazhenii, v. 1, Klassifikatsiya kriticheskikh tochek kaustik i volnovykh frontov, Nauka, M., 1982

[2] Arkhipov G. I., Karatsuba A. A., Chubarikov V. N., “Trigonometricheskie integraly”, Izv. AN SSSR, 43:5 (1979), 971–1003 | MR

[3] Varchenko A. N., “Mnogogranniki Nyutona i otsenki ostsilliruyuschikh integralov”, Funkts. analiz i ego prilozh., 10:3 (1976), 13–38 | MR

[4] Vinogradov I. M., Metod trigonometricheskikh summ v teorii chisel, Nauka, M., 1980

[5] Ikromov I. A., “Ob otsenkakh preobrazovaniya Fure indikatora nevypuklykh mnozhestv”, Dokl. RAN, 331:3 (1993), 272–274 | Zbl

[6] Ikromov I. A., “Ob otsenkakh preobrazovaniya Fure indikatora nevypuklykh oblastei”, Funkts. analiz i ego prilozh., 29:3 (1995), 16–24 | MR | Zbl

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

[8] Ikromov I. A., “Summiruemost ostsillyatornykh integralov po parametram i problema ob ogranichenii preobrazovaniya Fure na krivykh”, Mat. zametki, 87:5 (2010), 734–755 | MR | Zbl

[9] Karpushkin V. N., “Teorema o ravnomernoi otsenke ostsilliruyuschikh integralov s fazoi, zavisyaschei ot dvukh peremennykh”, Trudy sem. im. I. G. Petrovskogo, 10, 1984, 150–169 | Zbl

[10] Palamodov V. P., “Obobschennye funktsii i garmonicheskii analiz”, Sovrem. probl. mat. Fundam. napravl., 72, 1991, 5–134 | MR

[11] Popov D. A., “Otsenki s konstantami dlya nekotorykh klassov ostsilliruyuschikh integralov”, Usp. mat. nauk, 52:1 (1997), 77–148 | MR | Zbl

[12] Sadullaev A. S., “Kriterii algebraichnosti analiticheskikh mnozhestv”, Funkts. analiz i ego prilozh., 6:1 (1972), 85–86 | MR | Zbl

[13] Khermander L., Analiz lineinykh differentsialnykh operatorov s chastnymi proizvodnymi, v. 1, Mir, M., 1986

[14] Chakhkiev M., Otsenki ostsilliruyuschikh integralov s vypukloi fazoi i ikh prilozheniya, Avtoreferat diss. d.f.-m.n., MGU, M., 2006

[15] Bak J.-G., Oberlin D., Seeger A., Restriction of Fourier transform to curves: An endpoint estimate with affine arclength measure, 2012, arXiv: 1109.1300v2 [math.CA]

[16] Duistermaat J., “Oscillatory integrals, Lagrange immersions and unifoldings of singularities”, Commun. Pure Appl. Math., 27:2 (1974), 207–281 | DOI | Zbl

[17] Erdös L'., Salmhofer M., “Decay of the Fourier transform of surfaces with vanishing curvature”, Math. Z., 257:2 (2007), 261–294 | DOI | MR | Zbl

[18] Hironaka H., “Resolution of singularities of an algebraic variety over a field of characteristic zero: I, II”, Ann. Math., 79 (1964), 109–326 | DOI

[19] Hua L.-K., “On the number of solutions of Tarry's problem”, Acta Sci. Sinica, 1:1 (1952), 1–76 | MR

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

[21] Ikromov I. A., Müller D., “Uniform estimates for the Fourier transform of surface carried measures in $\Bbb R^3$ and an application to Fourier restriction”, J. Fourier Anal. Appl., 17:6 (2011), 1292–1332 | DOI | MR | Zbl

[22] Ikromov I. A., Müller D., Fourier restriction for hypersurfaces in three dimensions and Newton polyhedra, Princeton University Press, Princeton—Oxford, 2016 | Zbl

[23] Mokenhaupt G., Bounds in Lebesgue spaces of oscillatory integral operators, Habilitationsschrift, Universität Siegen, 1996

[24] Phong D. H., Stein E. M., Sturm J. A., “On the growth and stability of real-analytic functions”, Am. J. Math., 121:3 (1999), 519–554 | DOI | Zbl

[25] Randol B., “On the asymptotic behavior of the Fourier transform of the indicator function of a convex set”, Trans. AMS, 139 (1970), 278–285

[26] Sadullaev A., “On Weierstrass polynomials”, Ann. Pol. Mat., 123 (2019), 473–479 | Zbl

[27] Sadullaev A., Ikromov I. A., “Oscillatory integrals and Weierstrass polynomials”, Bull. Nat. Univ. Uzbekistan. Math. Nat. Sci., 2:2 (2019), 125–139

[28] Sogge C. D., Fourier integrals in classical analysis, Cambridge Univ. Press, Cambridge, 1993 | Zbl

[29] 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

[30] Stein E. M., Harmonic analysis: real-valued methods, orthogonality and oscillatory integrals, Princeton University Press, Princeton, 1993

[31] Svensson I., “Estimates for the Fourier transform of the characteristic function of a convex set”, Ark. Mat., 9:1 (1970), 11–22

[32] Van der Corput J. G., “Zur Methode der stationaren phase. I”, Composito Math., 1 (1934), 15–38