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

TL;DR

This paper proves that locally p-admissible measures on the real line are characterized by local Muckenhoupt A_p weights, linking local and global measure properties through advanced analysis tools.

Contribution

It establishes that locally p-admissible measures on  are exactly those derived from local A_p weights, extending global characterizations to local settings.

Findings

Locally p-admissible measures on  are from local A_p weights.

The class of locally p-admissible weights is invariant under addition.

Measures p-admissible on an interval can be extended by reflection to global p-admissible measures.

Abstract

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 Bj\"orn, Buckley and Keith, together with tools from analysis in metric spaces, more specifically preservation of the doubling condition and Poincar\'e 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.