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Locally $p$-admissible measures on $\mathbb{R}$

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.