Admissibility versus $A_p$-conditions on regular trees

TL;DR

This paper demonstrates that on rooted K-ary trees, the combination of doubling and the $(1,p)$-Poincare inequality is equivalent to an $A_p$-condition, linking geometric and analytic properties.

Contribution

It establishes an equivalence between geometric and analytic conditions on regular trees, connecting doubling, Poincare inequalities, and $A_p$-conditions.

Findings

Doubling and Poincare inequality are equivalent to $A_p$-condition on regular trees.

Provides a characterization of analytic conditions via geometric properties.

Links between different conditions deepen understanding of analysis on trees.

Abstract

We show that the combination of doubling and $(1,p)$-Poincare inequality is equivalent to a version of the $A_p$-condition on rooted K-ary trees.