The capacity comparison conjecture for the fractional Riesz transform

Let $d\geq 2$, let $0<s<d$ with $s\notin\mathbf{N}$, and let $E\subset\mathbf{R}^d$ be compact. For a gauge $\Phi$, let $\operatorname{cap}_{\Phi,s}(E)$ denote the associated nonlinear capacity, and let

$$\gamma_s(E)=\sup\{\mu(E):\mu\in\mathcal{M}^+(\mathbf{R}^d),\ \operatorname{supp}(\mu)\subset E,\ \|R(\mu)\|_{L^{\infty}}\leq 1\}$$

be the $s$-dimensional Calderón–Zygmund capacity, where $R$ is the $s$-dimensional Riesz transform. Capacity comparison conjecture. There exist positive constants $A_1$ and $A_2$, depending on $s$ and $d$, such that

$$A_1\operatorname{cap}_{\Phi,s}(E)\leq\gamma_s(E)\leq A_2\operatorname{cap}_{\Phi,s}(E),$$

for every compact set $E\subset\mathbf{R}^d$, with $\Phi(t)=t^2$. This conjecture proposes two-sided comparability between the Calderón–Zygmund capacity and the nonlinear capacity associated with the quadratic gauge. The source gives no resolution of the conjecture.