Uniform estimates for oscillatory integrals with homogeneous polynomial phases of degree 4

TL;DR

This paper establishes sharp uniform estimates for oscillatory integrals with degree 4 homogeneous polynomial phases, extending previous results to less smooth functions.

Contribution

It provides the first sharp uniform estimates for such integrals with degree 4 polynomial phases, generalizing Karpushkin's theorem.

Findings

Sharp uniform estimates derived for degree 4 polynomial phases

Extension of Karpushkin's theorem to less smooth functions

Results applicable to a broader class of oscillatory integrals

Abstract

In this paper we consider the uniform estimates for oscillatory integrals with a two-order homogeneous polynomial phase. The estimate is sharp and the result is an analogue of the more general theorem of V. N. Karpushkin \cite{Karpushkin1983} for sufficiently smooth functions.