arXiv subjects

Ziyi He

Publications and source records attributed to Ziyi He.

21 records · Page 2Linked to original sources

arXiv2019

Real-Variable Characterizations of Local Hardy Spaces on Spaces of Homogeneous Type

Suppose that $(X,d,μ)$ is a space of homogeneous type, with upper dimension $μ$, in the sense of R. R. Coifman and G. Weiss. Let $η$ be the Hölder regularity index of wavelets constructed by P. Auscher and T. Hytönen. In this article, the authors introduce the local Hardy space $h^{*,p}(X)$ via local grand maximal functions and also characterize $h^{*,p}(X)$ via local radial maximal functions, local non-tangential maximal functions, locally atoms and local Littlewood--Paley functions. Furthermore, the authors establish the relationship between the global and the local Hardy spaces. Finally, the authors also obtain the finite atomic characterizations of $h^{*,p}(X)$. As an application, the authors give the dual spaces of $h^{*,p}(X)$ when $p\in(ω/(ω+η),1)$, which further completes the result of G. Dafni and H. Yue on the dual space of $h^{*,1}(X)$. This article also answers the question of R. R. Coifman and G. Weiss on the nonnecessity of any additional geometric assumptions except the doubling condition for the radial maximal function characterization of $H^1_{\mathrm{cw}}(X)$ when $μ(X)<\infty$.

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arXiv2018

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. In this article, motivated by the breakthrough work of P. Auscher and T. Hytönen 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 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.

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arXiv2018

A Complete Real-Variable Theory of Hardy Spaces on Spaces of Homogeneous Type

Let $(X,d,μ)$ be a space of homogeneous type, with the upper dimension $ω$, in the sense of R. R. Coifman and G. Weiss. Assume that $η$ is the smoothness index of the wavelets on $X$ constructed by P. Auscher and T. Hytönen. In this article, when $p\in(ω/(ω+η),1]$, for the atomic Hardy spaces $H_{\mathrm{cw}}^p(X)$ introduced by Coifman and Weiss, the authors establish their various real-variable characterizations, respectively, in terms of the grand maximal function, the radial maximal function, the non-tangential maximal functions, the various Littlewood-Paley functions and wavelet functions. This completely answers the question of R. R. Coifman and G. Weiss by showing that no any additional (geometrical) condition is necessary to guarantee the radial maximal function characterization of $H_{\mathrm{cw}}^1(X)$ and even of $H_{\mathrm{cw}}^p(X)$ with $p$ as above. As applications, the authors obtain the finite atomic characterizations of $H^p_{\mathrm{cw}}(X)$, which further induce some criteria for the boundedness of sublinear operators on $H^p_{\mathrm{cw}}(X)$. Compared with the known results, the novelty of this article is that $μ$ is not assumed to satisfy the reverse doubling condition and $d$ is only a quasi-metric, moreover, the range $p\in(ω/(ω+η),1]$ is natural and optimal.

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