Four theorems on uniform estimates of oscillatory integrals

Abstract

Exact order uniform estimates of oscillatory integrals with monomial phase are obtained. These results are close to the hypothesis of V. I. Arnold on uniform estimates of oscillatory integrals. Namely, for absolute values of oscillatory integrals, estimates of order ρ−1/klnn−1τ uniform with respect to phase and amplitude are derived for every sufficiently small perturbation of phase (i.e., the monomial \(x_1^{m_1 } \ldots x_n^{m_n }\), mj ≤ k, 1 ≤ k, is perturbed by monomials \(x_1^{s_1 } \ldots x_n^{s_n }\), where sj ≤ k, 1 ≤ j ≤ n), and for each amplitude φ ∈ C02(Rn), n > 0. In the case |m| < nk the upper uniform estimate with the same perturbation and the same amplitude has the order τ−1/klnn−2τ. The estimate by order τ−1/klnn−2τ was proved in the case when the amplitude vanishes at the origin. In the case k = 1, a uniform estimate of order τ−1lnn−2τ is valid. This implies a uniform estimate for a polynomial phase. Previously the upper estimate of an oscillatory integral by the value 32nτ−1/klnn−1(τ+2) was known for the amplitude being the characteristic function of a cube and the same phase.