Orthonormal bases of regular wavelets in spaces of homogeneous type

TL;DR

This paper constructs orthonormal wavelet bases with H"older regularity and exponential decay in spaces of homogeneous type, enabling advanced analysis without extra space conditions.

Contribution

It introduces spline functions in geometrically doubling quasi-metric spaces and builds wavelet bases with desirable properties, extending classical wavelet theory to broader spaces.

Findings

Wavelet bases with exponential decay are constructed.

The approach applies to various function spaces like L^p and BMO.

The method proves the T(1) theorem without additional space conditions.

Abstract

Adapting the recently developed randomized dyadic structures, we introduce the notion of spline function in geometrically doubling quasi-metric spaces. Such functions have interpolation and reproducing properties as the linear splines in Euclidean spaces. They also have H\"older regularity. This is used to build an orthonormal basis of H\"older-continuous wavelets with exponential decay in any space of homogeneous type. As in the classical theory, wavelet bases provide a universal Calder\'on reproducing formula to study and develop function space theory and singular integrals. We discuss the examples of $L^p$ spaces, BMO and apply this to a proof of the T(1) theorem. As no extra condition {(like 'reverse doubling', 'small boundary' of balls, etc.)} on the space of homogeneous type is required, our results extend a long line of works on the subject.