$p$-Poincaré inequalities and cutoff Sobolev inequalities on metric measure spaces

Abstract

For $p>1$, 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 $\Phi$, and that both the Poincaré inequality and the the cutoff Sobolev inequality 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 there exists a metric measure space which 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 [Ann. Probab., to appear].

50 pages. Minor revision: a small gap in the proof of Proposition 8.9 has been fixed, and the references have been updated