Four theorems on uniform estimates of oscillatory integrals

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 $\tau^{-1/k}\ln^{n-1}\tau$ uniform with respect to phase and amplitude for every sufficiently small perturbation phase, i.e., the monomial $x_1^{m_1}\ldots x_n^{m_n}, m_j\leq k, 1\leq k$, by monomials $x_1^{s_1}\ldots x_n^{s_n}$, where $s_j\leq k, 1\leq j\leq n$, and for each amplitude $\varphi\in C_0^2(R^n), n>0$. In the case $|m|<nk$ the upper uniform estimate with the same perturbation and the same amplitude have the order $\tau^{-1/k}\ln^{n-2}\tau.$ The estimate by order $\tau^{-1/k}\ln^{n-2}\tau$ was proved in the case when the amplitude vanishes at the origin. In the case $k=1$, a uniform estimate of order $\tau^{-1}\ln^{n-2}\tau$ is valid. This implies a uniform estimate for a polynomial phase. The upper estimate of oscillatory integral $32^n\tau^{-1/k}\ln^{n-1}(\tau+2)$ was known previously for the amplitude being the characteristic function of a cube and the same phase.