Macroscopic Poincaré Inequalities and Weak-Type Characterizations of $p$-Energy Forms

For a class of regular strongly local $p$-energy forms on complete doubling metric measure spaces, $1<p<\mathrm{\infty }$, we derive a macroscopic Poincaré inequality from a Poincaré inequality and a capacity upper bound at a doubling scale $\mathrm{\Psi }$. The macroscopic inequality compares $L^p$-oscillations at any two scales and yields a uniform two-scale comparison $J_s\lesssim J_r$, $0<r<s$, for the Korevaar-Schoen functionals, strengthening their usual weak monotonicity property. Conversely, under a density assumption on compactly supported functions, the macroscopic inequality yields a regular strongly local Korevaar-Schoen $p$-energy satisfying a $\mathrm{\Psi }$-Poincaré inequality. We also show that the macroscopic inequality forces lower $p$-scaling of $\mathrm{\Psi }$ on scales not exceeding $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}X$; together with doubling, this gives two-sided power bounds. Finally, we establish a weak-type characterization of the energy domain of Brezis-Van Schaftingen-Yung type.