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

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 $\mathrm{\Phi }$, and that both the Poincaré inequality and the the cutoff Sobolev inequality with respect to a doubling scaling function $\mathrm{\Psi }$ hold if and only if $$\frac{1}{C}{\left(\frac{R}{r}\right)}^p\le \frac{\mathrm{\Psi }(R)}{\mathrm{\Psi }(r)}\le C{\left(\frac{R}{r}\right)}^{p-1}\frac{\mathrm{\Phi }(R)}{\mathrm{\Phi }(r)}\text{ for any }r\le R\mathrm{.}$$ In particular, given any pair of doubling functions $\mathrm{\Phi }$ and $\mathrm{\Psi }$ 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 $$p\le {\beta }_p\le d_h+(p-1)\mathrm{.}$$ Our proof builds on the Laakso-type space theory, which was recently developed by Murugan [Ann. Probab., to appear].