$p$-Poincar\'e inequalities and capacity upper bounds on metric measure spaces

For $p\in(1,+\infty)$, we prove that a metric measure space endowed with a $p$-energy satisfies the chain condition, the volume regular condition with respect to a doubling scaling function $\Phi$, and that both the Poincar\'e inequality and the capacity upper bound with respect to a doubling scaling function $\Psi$ hold if and only if $$\frac{1}{C}\left(\frac{R}{r}\right)^p\le\frac{\Psi(R)}{\Psi(r)}\le C\left(\frac{R}{r}\right)^{p-1}\frac{\Phi(R)}{\Phi(r)}\text{ for any }r\le R.$$ In particular, given any pair of doubling functions $\Phi$ and $\Psi$ 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 a metric measure space is $d_h$-Ahlfors regular and has $p$-walk dimension $\beta_p$ if and only if $$p\le\beta_p\le d_h+(p-1).$$ Our proof builds on the Laakso-type space theory, which was recently developed by Murugan (arXiv:2410.15611).