Almost Lipschitz-continuous wavelets in metric spaces via a new randomization of dyadic cubes

TL;DR

This paper improves the construction of wavelets in metric spaces, allowing for nearly Lipschitz-continuous wavelets by enhancing the randomization of dyadic cubes, thus broadening their regularity properties.

Contribution

It introduces a new randomization method for dyadic cubes that achieves wavelet H"older-exponents arbitrarily close to 1 in metric spaces.

Findings

Wavelet H"older-exponent can be made arbitrarily close to 1

Improved random dyadic cube construction with broader applications

Enhanced wavelet regularity in metric spaces

Abstract

In any quasi-metric space of homogeneous type, Auscher and Hyt\"onen recently gave a construction of orthonormal wavelets with H\"older-continuity exponent $\eta>0$. However, even in a metric space, their exponent is in general quite small. In this paper, we show that the H\"older-exponent can be taken arbitrarily close to 1 in a metric space. We do so by revisiting and improving the underlying construction of random dyadic cubes, which also has other applications.