On oscillatory integrals with H\"older phases

TL;DR

This paper constructs a family of fractal-like H"older maps that satisfy a Van Der Corput type lemma, providing a simpler proof and improved decay control for oscillatory integrals with fractal phases.

Contribution

It introduces a new elementary approach to establish decay estimates for oscillatory integrals with fractal phases, simplifying previous proofs and enhancing decay rate control.

Findings

Constructed autosimilar H"older maps satisfying a fractal Van Der Corput lemma

Provided a simpler proof based on elementary non-concentration estimates

Achieved better control over decay rates in oscillatory integrals with fractal phases

Abstract

We exhibit a family of autosimilar H\"older maps that satisfies a fractal version of the Van Der Corput Lemma, despite not being absolutely continuous. The result is a direct consequence of a recent work of Sahlsten and Steven arXiv:2009.01703, which is based on a powerful theorem of Bourgain known as a sum-product phenomenon estimate. We give a substantially simpler proof of this fact in our particular context, using an elementary method inspired from arXiv:1704.02909 to check the non-concentration estimates that are needed to apply the sum-product phenomenon. This method allows us to gain additional control over the decay rate.