Uniform estimates for oscillatory integrals with smooth phase

TL;DR

This paper establishes sharp uniform estimates for oscillatory integrals with smooth phase functions exhibiting singularities, extending the analogy of previous work by V.N. Karpushkin.

Contribution

It provides new sharp uniform estimates for oscillatory integrals with singular smooth phases, advancing the understanding of their behavior.

Findings

Establishes sharp uniform bounds for oscillatory integrals with singular phases.

Extends previous estimates to broader classes of phase functions.

Provides a theoretical framework analogous to Karpushkin's work.

Abstract

We consider the problem on uniform estimates for an oscillatory integrals with the smooth phase functions having singularities $D_{\infty} $. The estimate is sharp and analogy to estimates of the work of V.N.Karpushkin.