New Calder\'on Reproducing Formulae with Exponential Decay on Spaces of Homogeneous Type

Abstract

Assume that $(X, d, \mu)$ is a space of homogeneous type in the sense of Coifman and Weiss. In this article, motivated by the breakthrough work of P. Auscher and T. Hyt\"onen on orthonormal bases of regular wavelets on spaces of homogeneous type, the authors introduce a new kind of approximations of the identity with exponential decay (for short, $\exp$-ATI). Via such an $\exp$-ATI, motivated by another creative idea of Y. Han et al. to merge the aforementioned orthonormal bases of regular wavelets into the frame of the existed distributional theory on spaces of homogeneous type, the authors establish the homogeneous continuous/discrete Calder\'on reproducing formulae on $(X, d, \mu)$, as well as their inhomogeneous counterparts. The novelty of this article exists in that $d$ is only assumed to be a quasi-metric and the underlying measure $\mu$ a doubling measure, not necessary to satisfy the reverse doubling condition. It is well known that Calder\'on reproducing formulae are the cornerstone to develop analysis and, especially, harmonic analysis on spaces of homogeneous type.