Geodesic
Remarks on uniform combined estimates of oscillatory integrals
Izvestiya. Mathematics , Tome 72 (2008) no. 4, pp. 793-816.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the problem of constructing asymptotically exact (for $\Omega\gg 1$) uniform (with respect to parameters $t=(t_1,t_2,\dots,t_m)$) estimates for oscillatory integrals containing a large parameter $\Omega$. We suggest a possible multidimensional analogue of Vinogradov's well-known estimate for one-dimensional integrals. Based on this suggestion, we estimate the integrals with singularities of type $A_k$, $D_4^{\pm}$ (in Arnold's classification) and use the special case of $D_5^\pm$ to discuss the possibility of generalizing our results. 
@article{IM2_2008_72_4_a7,
     author = {D. A. Popov},
     title = {Remarks on uniform combined estimates of oscillatory integrals},
     journal = {Izvestiya. Mathematics },
     pages = {793--816},
     publisher = {mathdoc},
     volume = {72},
     number = {4},
     year = {2008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2008_72_4_a7/}
}
TY  - JOUR
AU  - D. A. Popov
TI  - Remarks on uniform combined estimates of oscillatory integrals
JO  - Izvestiya. Mathematics 
PY  - 2008
SP  - 793
EP  - 816
VL  - 72
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2008_72_4_a7/
LA  - en
ID  - IM2_2008_72_4_a7
ER  - 
%0 Journal Article
%A D. A. Popov
%T Remarks on uniform combined estimates of oscillatory integrals
%J Izvestiya. Mathematics 
%D 2008
%P 793-816
%V 72
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2008_72_4_a7/
%G en
%F IM2_2008_72_4_a7
D. A. Popov. Remarks on uniform combined estimates of oscillatory integrals. Izvestiya. Mathematics , Tome 72 (2008) no. 4, pp. 793-816. http://geodesic.mathdoc.fr/item/IM2_2008_72_4_a7/

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

[2] L. Hörmander, The analysis of linear partial differential operators. V. I. Distribution theory and Fourier analysis, Grundlehren der Mathematischen Wissenschaften, 256, Springer-Verlag, Berlin, 1983 | MR | MR | Zbl | Zbl

[3] V. I. Arnol'd, “Remarks on the stationary phase method and Coxeter numbers”, Russ. Math. Surveys, 28:5 (1973), 19–48 | DOI | MR | Zbl | Zbl

[4] V. I. Arnol'd, S. M. Gusein-Zade, A. N. Varchenko, Singularities of differentiable maps. II. Monodromy and the asymptotic behavior of integrals, Birkhäuser, Boston, 1988 | MR | Zbl | Zbl

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

[6] V. Guillemin, Sh. Stenberg, Geometric asymptotics, Mathematical Surveys, 14, Amer. Math. Soc., Providence, RI, 1977 | MR | MR | Zbl | Zbl

[7] V. P. Maslov and M. V. Fedoryuk, Semi-classical approximation in quantum mechanics, Nauka, Moscow, 1976 | MR | Zbl | Zbl

[8] Reidel, Dordrecht, 1981 | MR | Zbl | Zbl

[9] I. M. Vinogradov, The method of trigonometric sums in the theory of numbers, Dover Publ., Mineola, NY, 2004 | MR | Zbl | Zbl

[10] I. M. Vinogradov, Osobye varianty metoda trigonometricheskikh summ, Nauka, M., 1976 | MR | Zbl

[11] J. G. van der Corput, “Neue Zahlentheoretische Abschatzungen”, Math. Ann., 89 (1923), 215–254 | DOI | Zbl

[12] G. I. Arkhipov, A. A. Karatsuba, V. N. Chubarikov, “Trigonometric integrals”, Math. USSR Izv., 15:2 (1980), 211–239 | DOI | MR | Zbl

[13] G. I. Arkhipov, A. A. Karatsuba, V. N. Chubarikov, Teoriya kratnykh trigonometricheskikh summ, Nauka, M., 1987 | MR | Zbl

[14] G. I Arkhipov, V. N. Chubarikov, A. A. Karatsuba, Trigonometric sums in number theory and analysis, de Gruyter Expositions in Mathematics, 39, Walter de Gruyter, Berlin, 2004 | MR | Zbl

[15] D. V. Kosygin, A. A. Minasov, Ya. G. Sinai, “Statistical properties of the spectra of Laplace–Beltrami operators on Liouville surfaces”, Russian Math. Surveys, 48:4 (1993), 1–142 | DOI | MR | Zbl

[16] J. Bourgain, “$L^p$-estimaties for oscillatory integrals in sevral variables”, Geom. Funct. Anal., 1:4 (1991), 321–374 | DOI | MR | Zbl

[17] E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, 43, Princeton Univ. Press, Princeton, NJ, 1993 | MR | Zbl

[18] D. A. Popov, “Spherical convergence of the Fourier series and integral of the indicator of a two-dimensional domain”, Proc. Steklov Inst. Math., 218:3 (1997), 352–371 | MR | Zbl

[19] D. A. Popov, “Estimates with constants for some classes of oscillatory integrals”, Russian Math. Surveys, 52:1 (1997), 73–145 | DOI | MR | Zbl

[20] D. A. Popov, “Reconstruction of characteristic functions in two-dimensional Radon tomography”, Russian Math. Surveys, 53:1 (1998), 109–193 | DOI | MR | Zbl

[21] V. I. Arnol'd, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of differentiable maps. I. Classification of critical points, caustics and wave fronts, Birkhäuser, Boston, 1985 | MR | Zbl | Zbl

[22] V. I. Arnol'd, “Normal forms for functions near degenerate critical points, the Weyl groups of $A_k$, $D_k$, $E_k$ and Lagrangian singularities”, Funct. Anal. Appl., 6:4 (1972), 254–272 | DOI | MR | Zbl

[23] Y. Colin de Verdièr, “Nombre de points entiers dans une famille homothétique de domains de $\mathbb R^n$”, Ann. Sci. École Norm. Sup. (4), 10:4 (1977), 559–575 | MR

[24] V. N. Karpushkin, “Ravnomernye otsenki ostsilliruyuschikh integralov s giperbolicheskoi i parabolicheskoi fazoi”, Tr. semin. im. I. G. Petrovskogo, 1983, no. 9, 3–39 | MR | Zbl

[25] V. N. Karpushkin, “Teorema o ravnomernykh otsenkakh ostsilliruyuschikh integralov s fazoi, zavisyaschei ot dvukh peremennykh”, Tr. sem. im. I. G. Petrovskogo, 10 (1984), 150–169 | MR | Zbl

[26] V. N. Karpushkin, “Uniform estimates of volumes”, Proc. Steklov Inst. Math., 221 (1998), 214–220 | MR | Zbl

[27] A. N. Varchenko, “Newton polyhedra and estimation of oscillating integrals”, Funct. Anal. Appl., 10:3 (1976), 175–196 | DOI | MR | Zbl

[28] Th. Bröcker, Differentiable germs and catastrophes, London Mathematical Society Lecture Note Series, 17, Cambridge Univ. Press, 1975 | MR | Zbl

[29] G. V. Gurevich, Osnovy algebraicheskoi teorii invariantov, GITTL, M., 1948