$p$-Poincaré inequalities and cutoff Sobolev inequalities on metric measure spaces
Meng Yang
TL;DR
This work extends the interplay between Poincaré inequalities and cutoff Sobolev inequalities to general metric measure spaces carrying a $p$-energy, establishing a sharp two-sided condition relating two doubling scaling functions $\Phi$ and $\Psi$ under the chain condition. It proves that $PI(\Psi)$ and $CS(\Psi)$ hold together with a capacity bound $cap(\Psi)_{\le}$ if and only if $\frac{1}{C}\left(\frac{R}{r}\right)^p \le \frac{\Psi(R)}{\Psi(r)} \le C\left(\frac{R}{r}\right)^{p-1}\frac{\Phi(R)}{\Phi(r)}$ for all $r\le R$, and constructs, for any admissible $\Phi,\Psi$, a Laakso-type space supporting a $p$-energy with these properties. A key application is a precise characterization of when a metric measure space can be $d_h$-Ahlfors regular with $p$-walk dimension $\beta_p$, namely $p \le \beta_p \le d_h+(p-1)$. The construction combines $\mathbb{R}$-trees and ultrametric spaces into Laakso-type spaces, employing the pencil-of-curves method to obtain PI and CS, and leverages potential-theoretic foundations to justify measurability and energy arguments. The results provide a concrete geometric realization of the admissible $\Phi$–$\Psi$ pairs and yield sharp Besov-space critical exponent ranges in this context.
Abstract
For $p\in(1,+\infty)$, we introduce the cutoff Sobolev inequality on general metric measure spaces, and prove that there exists a metric measure space endowed with a $p$-energy that satisfies the chain condition, the volume regular condition with respect to a doubling scaling function $Φ$, and that both the Poincaré inequality and the the cutoff Sobolev inequality with respect to a doubling scaling function $Ψ$ hold if and only if $$\frac{1}{C}\left(\frac{R}{r}\right)^p\le\frac{Ψ(R)}{Ψ(r)}\le C\left(\frac{R}{r}\right)^{p-1}\frac{Φ(R)}{Φ(r)}\text{ for any }r\le R.$$ In particular, given any pair of doubling functions $Φ$ and $Ψ$ satisfying the above inequality, we construct a metric measure space endowed with a $p$-energy on which all the above conditions are satisfied. As a direct corollary, we prove that there exists a metric measure space which is $d_h$-Ahlfors regular and has $p$-walk dimension $β_p$ if and only if $$p\leβ_p\le d_h+(p-1).$$ Our proof builds on the Laakso-type space theory, which was recently developed by Murugan (arXiv:2410.15611).