Geodesic
Randol Maximal Functions and the Integrability of the Fourier Transform of Measures
Matematičeskie zametki, Tome 109 (2021) no. 5, pp. 643-663.

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Estimates of the Fourier transform of charges (measures) concentrated on smooth hypersurfaces are considered. Following M. Sugumoto, three classes of smooth hypersurfaces are defined. Depending on the class, estimates of the Fourier transform of charges are obtained in terms of Randol maximal functions. The obtained estimates are applied to the solution of the integrability problem for the Fourier transform of measures concentrated on some nonconvex hypersurfaces. The sharpness of the obtained estimates is shown.
Keywords: measure, curvature, integrability.
Mots-clés : Fourier transform, hypersurface 
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D. I. Akramova; I. A. Ikromov. Randol Maximal Functions and the Integrability of the Fourier Transform of Measures. Matematičeskie zametki, Tome 109 (2021) no. 5, pp. 643-663. http://geodesic.mathdoc.fr/item/MZM_2021_109_5_a0/

[1] V. P. Palamodov, “Obobschennye funktsii i garmonicheskii analiz”, Kommutativnyi garmonicheskii analiz – 3, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Fundam. napravleniya, 72, VINITI, M., 1991, 5–134 | MR | Zbl

[2] V. V. Lebedev, “O preobrazovanii Fure kharakteristicheskikh funktsii oblastei s $C^1$-gladkoi granitsei”, Funkts. analiz i ego pril., 47:1 (2013), 33–46 | DOI | MR | Zbl

[3] E. M. Stein, Harmonic Analysis: Real-Valued Methods, Orthogonality and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993 | MR

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

[5] I. M. Vinogradov, Metod trigonometricheskikh summ v teorii chisel, Nauka, M., 1971 | MR | Zbl

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

[7] I. A. Ikromov, D. Müller, Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra, Ann. of Math. Stud., 194, Princeton Univ. Press, Princeton, NJ, 2016 | MR | Zbl

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

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

[10] M. Sugimoto, “Estimates fo Hyperbolic Equations of Space Dimension 3”, J. Funct. Anal., 160:2 (1998), 382–407 | DOI | MR | Zbl

[11] M. V. Fedoryuk, Metod perevala, Nauka, M., 1977 | MR | Zbl

[12] B. A. Dubrovin, S. P. Novikov, A. T. Fomenko, Sovremennaya geometriya. Metody i prilozheniya, Nauka, M., 1979 | MR | Zbl

[13] V. I. Arnold, A. N. Varchenko, S. M. Gusein-zade, “Osobennosti differentsiruemykh otobrazhenii”, Ch. 1, Klassifikatsiya kriticheskikh tochek kaustik i volnovykh frontov, Nauka, M., 1982

[14] B. Randol, “On the asymptotic behavior of the Fourier transform of the indicator function of a convex set”, Trans. Amer. Math. Soc., 139 (1970), 271–278 | MR

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

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

[17] I. A. Ikromov, “Summiruemost preobrazovaniya Fure mer, sosredotochennykh na giperpoverkhnostyakh”, Funkts. analiz i ego pril., 49:3 (2015), 74–79 | DOI | Zbl

[18] V. P. Maslov, Teoriya vozmuschenii i asimptoticheskie metody, Izd-vo Mosk. un-ta, M., 1965

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

[20] A. N. Varchenko, “O chisle tselykh tochek v semeistvakh gomotetichnykh oblastei v $\mathbb{R}^n$”, Funkts. analiz i ego pril., 17:2 (1983), 1–6 | MR | Zbl

[21] L. Khermander, Analiz lineinykh differentsialrykh operatorov s chastnymi proizvodnymi. T. 1. Teoriya raspredelenii i analiz Fure, Mir, M., 1986 | MR | Zbl

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