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

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 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.

arXiv comment13 pages

References in corpus (4)

• Local and semilocal Poincaré inequalities on metric spaces
• Poincaré inequalities and Newtonian Sobolev functions on noncomplete metric spaces
• Sharp capacitary estimates for rings in metric spaces
• Local behavior of p-harmonic Green's functions in metric spaces

Cited by in corpus (4)

• The Liouville theorem for $p$-harmonic functions and quasiminimizers with finite energy
• Removable sets for Newtonian Sobolev spaces and a characterization of $p$-path almost open sets
• Classification of metric measure spaces and their ends using $p$-harmonic functions
• Bounded geometry and $p$-harmonic functions under uniformization and hyperbolization