Four theorems on uniform estimates for oscillating integrals / *V. N. Karpushkin.* // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2014. № 3. P. 56-60
[Moscow Univ. Math. Bulletin. Vol. 72, N 2, 2017. P. 0].

The exact order of uniform estimates of oscillatory integrals with
monomial phase is obtaind. This result is close to the hypothesis
of V. I. Arnold about uniform estimates of oscillatory integrals.
Namely, for absolute values of oscillatory integrals we derive estimates of
order τ^{-1/k} ln^{n-1}τ uniform with respect to phase and amplitude
for every sufficiently small perturbation phase, i.e., the monomial
*x*_{1}^{m1}…*x _{n}^{mn}*, by monomials

*x*

_{1}

^{s1}…

*x*, where

_{n}^{sn}*s*≤ k, 1 ≤

_{j}*j ≤ n*, and for each amplitude φ ∈

*C*

_{0}

^{2}(ℝ

^{n}),

*n*>0. In the case |

*m*| <

*nk*the upper uniform estimate with the same perturbation and the same amplitude have the order τ

^{-1/k}ln

^{n-2}τ The estimate by order τ

^{-1/k}ln

^{n-2}τ was proved in the case when the amplitude vanishes at the origin. In the case

*k*= 1, a uniform estimate of order τ

^{-1}ln

^{n-2}τ is valid. This implies a uniform estimate for a polynomial phase. The upper estimate of oscillatory integral 32

^{n}τ

^{-1/k}ln

^{n-1}(τ+2) was known previously for the amplitude being the characteristic function of a cube and the same phase.

*Key words*:
oscillatory integral, phase, amplitude, uniform estimate.