Let X be a metric space equipped with a measure satisfying the doubling and reverse doubling conditions. In this paper, we develop the theory of new localized Hardy spaces \(H^p_\rho (X)\) for \(\frac{n}{n+1}<p\le 1\) associated to critical functions \(\rho \) defined on X where n is the doubling order. Our results include the atomic decomposition characterization and the maximal function characterization associated to an approximation of the identity. We then study the T1 criteria for singular integrals to be bounded on our Hardy spaces.

中文翻译：

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与临界函数相关的Hardy空间及其对T 1定理的应用

令X为配备有满足加倍和反向加倍条件的度量的度量空间。在本文中，我们针对与关键函数相关的\（\ frac {n} {n + 1} <p \ le 1 \）开发了新的局部Hardy空间\（H ^ p_ \ rho（X）\）的理论。 （\ rho \）在X上定义，其中n是加倍顺序。我们的结果包括与恒等式近似相关的原子分解表征和最大功能表征。然后，我们研究要在Hardy空间上有界的奇异积分的T 1准则。