A theorem concerning uniform estimates of oscillatory integrals when the phase is a function of two variables

In the article it is shown that an oscillatory integral with phase depending on two variables has an upper estimate for small parameters which is proportional to the value of the integral when the parameters are zero. An analogous uniform estimate is obtained for volumes. V. I. Arnold's conjecture about the semicontinuity of the height for functions of two variables is also proved.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 10, pp. 150–169, 1984.The author is grateful to V. I. Arnol'd for posing the problem and for his advice with the work, and also to A. N. Varchenko for numerous valuable discussions.