$\mathbb{R}$上的局部$p$ -容许测度
Locally $p$-admissible measures on $\mathbb{R}$
摘要：
本文证明了$\mathbb{R}$上的局部$p$ -容许测度必然来自局部Muckenhoupt $A_p$权.在证明中，我们采用了相应的全球$p$ -容许措施$\mathbb{R}$在全球$A_p$权重由于Björn，巴克利和基思，与工具，从分析度量空间，更具体地说，保持加倍条件和庞加莱不等式平坦下，由于Durand-Cartagena和李。因此，$\mathbb{R}$上的局部$p$ -容许权类在加法下不变，并且满足格性质。我们还表明，措施是$p$ -容许的间隔可以部分扩展的周期性反射全球$p$ -容许措施。令人惊讶的是，$p$ -容许性必须保持在比反射的更大的间隔上，并且一个例子表明这是必要的。
英文摘要：
In this note we show that locally $p$-admissible measures on $\mathbb{R}$ necessarily come from local Muckenhoupt $A_p$ weights. In the proof we employ the corresponding characterization of global $p$-admissible measures on $\mathbb{R}$ in terms of global $A_p$ weights due to Björn, Buckley and Keith, together with tools from analysis in metric spaces, more specifically preservation of the doubling condition and Poincaré inequalities under flattening, due to Durand-Cartagena and Li. As a consequence, the class of locally $p$-admissible weights on $\mathbb{R}$ is invariant under addition and satisfies the lattice property. We also show that measures that are $p$-admissible on an interval can be partially extended by periodical reflections to global $p$-admissible measures. Surprisingly, the $p$-admissibility has to hold on a larger interval than the reflected one, and an example shows that this is necessary.
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