Multidimensional decay in van der Corput lemma

TL;DR

This paper extends the van der Corput lemma to multiple dimensions, allowing for decay estimates of oscillatory integrals across all variables, bridging one-dimensional results with the stationary phase method.

Contribution

It introduces a multidimensional version of the van der Corput lemma, enhancing decay analysis of oscillatory integrals in higher dimensions.

Findings

Established a multidimensional decay estimate for oscillatory integrals.

Connected the classical van der Corput lemma with the stationary phase method.

Provided a theoretical framework for analyzing multidimensional oscillatory behavior.

Abstract

In this paper we present a multidimensional version of the van der Corput lemma where the decay of the oscillatory integral is gained with respect to all space variables, connecting the standard one-dimensional van der Corput lemma with the stationary phase method.