New Calderón reproducing formulae with exponential decay on spaces of homogeneous type

Assume that(X,d,μ) is a space of homogeneous type in the sense of Coifman and Weiss(1971,1977). In this article, motivated by the breakthrough work of Auscher and Hyt(o|¨)nen(2013) on orthonormal bases of regular wavelets on spaces of homogeneous type, we 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 Han et al.(2018) to merge the aforementioned orthonormal bases of regular wavelets into the frame of the existed distributional theory on spaces of homogeneous type, we establish the homogeneous continuous/discrete Calderón reproducing formulae on(X, d,μ), 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 μ a doubling measure,not necessary to satisfy the reverse doubling condition. It is well known that Calderón reproducing formulae are the cornerstone to develop analysis and, especially, harmonic analysis on spaces of homogeneous type.

Atomic Decompositions of Triebel-Lizorkin Spaces with Local Weights and Applications

Abstract In this paper, the authors characterize the inhomogeneous Triebel-Lizorkin spaces Fs,w p,q （Rn） with local weight w by using the Lusin-area functions for the full ranges of the indices, and then establish their atomic decompositions for s ∈ R, p ∈ （0, 1] and q ∈ [p, ∞）. The novelty is that the weight w here satisfies the classical Muckenhoupt condition only on balls with their radii in （0, 1]. Finite atomic decompositions for smooth functions in Fs,w p,q（Rn） are also obtained, which further implies that a （sub）linear operator that maps smooth atoms of Fs,w p,q（Rn） uniformly into a bounded set of a （quasi-）Banach space is extended to a bounded operator on the whole Fs,w p,q（Rn） As an application, the baundedness of the local Riesz operator on the space Fs,w p,q（Rn） is obtained.