Scalar oscillatory integrals in smooth spaces of homogeneous type

TL;DR

This paper extends the theory of scalar oscillatory integrals to smooth spaces of homogeneous type with differentiability structures, providing uniform estimates that respect the underlying geometry and generalizing existing asymptotic results.

Contribution

It introduces a new framework for analyzing oscillatory integrals on generalized spaces of homogeneous type with differentiability, broadening the scope of classical estimates.

Findings

Established uniform estimates for scalar oscillatory integrals respecting geometric structures

Generalized a theorem on asymptotic behavior of oscillatory integrals with convex phases

Connected geometric analysis with oscillatory integral estimates in new spaces

Abstract

We consider a generalization of the notion of spaces of homogeneous type, inspired by recent work of Street [21] on the multi-parameter Carnot-Caratheodory geometry, which imbues such spaces with differentiability structure. The setting allows one to formulate estimates for scalar oscillatory integrals on these spaces which are uniform and respect the underlying geometry of both the space and the phase function. As a corollary we obtain a generalization of a theorem of Bruna, Nagel, and Wainger [1] on the asymptotic behavior of scalar oscillatory integrals with smooth, convex phase of finite type.