Mathematics & Physics 2016, 9(1), 102-107 УДК 517.518.5 On Invariant Estimates for Oscillatory Integrals with Polynomial Phase Akbar R. Safarov∗ Samarkand State University Universitetsky boulevard, 15 140104, Samarkand Uzbekistan Received 17.04.2015, received in revised form 10.11.2015, accepted 21.12.2015 In this paper we consider estimates for trigonometric (oscillatory) integrals with polynomial phase function of degree three. <...> The main result of the paper is the theorem on uniform invariant estimates for trigonometric integrals. <...> This estimate improves results obtained in the paper by D. A. Popov [1] for the case when the phase function is a sum of a homogeneous polynomial of third degree and a linear function, as well as the estimates of the paper [2] for the fundamental solution to the dispersion equation of third order. <...> Note that the form of such oscillatory integrals does not change under linear changes of variables, i.e. it is invariant under linear changes of variables. <...> V.P.Palamodov posed the problem of estimating the trigonometric integral in terms of the coefficients of the phase function [3]. <...> In this paper we give a solution of Palamodov’s problem when the phase function is a sum of a homogeneous polynomial of third degree and linear terms. <...> Such kind of integrals has been considered in the paper [2] in relation to the fundamental solution to third order dispersion equations. <...> However, in that paper the estimates were obtained for the fundamental solutions for fixed coefficients of the principal part. <...> It is an interesting problem to investigate the behavior of corresponding oscillatory integrals that depends on the coefficients of the polynomial, as proposed by V.P.Palamodov. <...> However, these estimates are not optimal when discriminant of P3 is small and the coefficients are large. <...> In this paper we obtain optimal invariant estimates for trigonometric integrals (see Theorem 2.1). <...> There is a well-known asymptotic expansion for trigonometric integrals with smooth amplitude function as coefficients of the phase tend to infinity along a fixed direction, say λA, where A ∈ S5 is a fixed point on the unit sphere centered at the origin and λ is a positive parameter. <...> However, the behavior of a trigonometric integral may change significantly even if the vector A varies a little. <...> Thus, we come to the problem of combined estimates for trigonometric <...>