Comparability of p-capacity to (d−4)-Bessel–Riesz capacity

Determine whether there exist constants c,C>0 depending only on dimension d such that for every finite set A ⊂ Z^d in the high-dimensional critical percolation regime, c·Cap_{d−4}(A) ≤ pCap(A) ≤ C·Cap_{d−4}(A), where Cap_{d−4}(A) denotes the (d−4)-Bessel–Riesz capacity defined by the variational formula Cap_{d−4}(A) = ( inf_{μ} ∑_{x,y∈A} (1+||y−x||)^{4−d} μ(x)μ(y) )^{-1} with μ ranging over probability measures on A.

Background

The authors prove a general lower bound pCap(A) ≥ c·Cap_{d−4}(A) and match the order for balls, but a full two-sided comparability for arbitrary finite sets remains unsettled.

Establishing comparability would align p-capacity with branching capacity results and cement its role as a percolation analogue of Bessel–Riesz capacities.