Elliptic Harnack inequality and Poincaré inequality for $p$-energies on metric measure spaces

For $p>1$ and a $p$-energy on a volume doubling metric measure space, we prove the Poincaré inequality under the elliptic Harnack inequality, two-sided capacity bounds, and additional geometric and analytic assumptions. Combining this with our previous results, we obtain, under the same structural assumptions, the equivalence between the elliptic Harnack inequality together with two-sided capacity bounds and the conjunction of the Poincaré inequality and the cutoff Sobolev inequality.