On estimates for oscillatory integrals with phase depending on parameters

Sh. A. Muranov

Ufa mathematical journal, Tome 11 (2019) no. 4, pp. 78-90

Résumé

We consider estimates for the Fourier transforms of measures supported on analytic hypersurfaces involving a damping factor. As a damper, we naturally take a power of the Gaussian curvature of the surface. It is known that if the exponent in this power is a sufficiently large positive number, then the Fourier transform of the corresponding measure has an optimal decay. C.D. Sogge and E.M. Stein formulated a problem on a minimal power of the Gaussian curvature ensuring an optimal decay for the Fourier transform. In the paper we resolve the problem by C.D. Sogge and E.M. Stein on an optimal decay for the Fourier transform with a damping factor for a particular class of families of analytic surfaces in the three-dimensional Euclidean space. We note that the power we provide is sharp not only for the families of analytic hypersurfaces but also for a fixed analytic hypersurface. The proof of main result is based on the methods of the theory of analytic functions, more precisely, on the statements like a preparation Weierstrass theorem. As D.M. Oberlin showed, similar statements fail for infinitely differentiable hypersurfaces.

Détail

Keywords: oscillating integrals, dumping factor, maximal operator.
Mots-clés : Fourier transform

Bibliographie

[1] C. D. Sogge, E. M. Stein, “Averages of functions over hypersurfaces in $\mathbb{R}^n$”, Invent. Math., 82:3 (1985), 543–556 | DOI | MR | Zbl

[2] D. M. Oberlin, “Oscillatory integrals with polynomial phase”, Math. Scand., 69:1 (1991), 45–56 | DOI | MR | Zbl

[3] V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko, Singularities of differentiable maps, v. 1, Modern Birkhäuser Classics, Classification of critical points, caustics and wave fronts, Birkhäuser, Boston, 2012 | MR | MR

[4] Sh. A. Muranov, “On estimates for oscillatory integrals with damping factor”, Uzbek Math. J., 4 (2018), 112–125 | DOI | MR

[5] I. A. Ikromov, Sh. A. Muranov, “Estimates of oscillatory integrals with a damping factor”, Math. Notes, 104:2 (2018), 218–230 | DOI | DOI | MR | Zbl

[6] B. A. Dubrovin, A. T. Fomenko, S. P. Novikov, Modern geometry methods and applications, v. I, Graduate Texts in Mathematics, 93, The geometry of surfaces, transformation groups, and fields, Springer-Verlag, New York etc. | MR

[7] M. V. Fedoryuk, Saddle-point method, Nauka, M., 1977 (in Russian)

[8] E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, Princeton, 1993 | MR | Zbl

[9] Math. USSR-Izv., 15:2 (1980), 211–239 | DOI | MR | Zbl

[10] J. G. VanDer Corput, “Zahlentheoretische Abschätzungen”, Math. Ann., 84 (1921), 53–79 (in German) | DOI | MR

[11] I. A. Ikromov, “Damped oscillatory integrals and maximal operators”, Math. Notes, 78:6 (2005), 773–790 | DOI | DOI | MR | Zbl

[12] A. S. Sadullaev, “Criteria for analytic sets to be algebraic”, Funct. Anal. Appl., 6:1 (1972), 78–79 | DOI | MR | Zbl

[13] Ed. Bierstone, P. D. Milman, “Arc-analytic functions”, Invent. Math., 101 (1990), 411–424 | DOI | MR | Zbl

[14] A. Erdélyi, Asymptotic expansions, Dover Publications, New York, 1956 | MR | Zbl

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

[16] B. Malgrange, Ideals of differentiable functions, Oxford University Press, London, 1966 | MR | Zbl

[17] W. Osgood, Lehrbuch der Funktionentheorie, v. II, Teubner, Leipzig, 1929 (in German)