Bounded-below criterion for the four-dimensional Drury–Arveson reciprocal problem

Determine whether J_R(f)=\int_{\mathbb B_4}|Rf(z)|^4|f(z)|^{-6}\,dv(z) is finite for every function f\in H_4^2 satisfying \inf_{z\in\mathbb B_4}|f(z)|\geq c>0; equivalently, determine whether such a lower bound implies 1/f\in H_4^2.

Background

For the four-dimensional Drury–Arveson space H_4^2, the paper proves that 1/f belongs to H_4^2 if and only if the quantity J_R(f)=\int_{\mathbb B_4}|Rf|^4/|f|^6\,dv is finite. Thus, the reciprocal problem is reduced to deciding whether boundedness of 1/f, equivalently a positive lower bound for |f|, forces this nonlinear integral condition.

The authors explicitly state that the slicing method does not completely solve the four-dimensional case and that the essential unresolved issue is whether a uniform estimate, or equivalently the finiteness of J_R(f), follows from the lower bound on f.