The cancellation conjecture for Riesz potentials on constrained function spaces

Let $d$ and $\ell$ be positive integers, let $S^{d-1}$ be the unit sphere in $\mathbb{R}^d$, and let $G(\ell,k)$ denote the Grassmannian of $k$-dimensional linear subspaces of $\mathbb{C}^{\ell}$. Let $\Omega:S^{d-1}\to G(\ell,k)$ be smooth, and define

$$\mathfrak{W}=\left\{g\in L_1(\mathbb{R}^d,\mathbb{C}^{\ell})\ \middle|\ \widehat{g}(\xi)\in\Omega\left(\frac{\xi}{|\xi|}\right)\text{ for every }\xi\in\mathbb{R}^d\setminus\{0\}\right\}.$$

For $\alpha\in(0,d)$, the cancellation conjecture. The Riesz potential $\operatorname{I}_{\alpha}$ maps $\mathfrak{W}$ to $L_{d/(d-\alpha)}$ if and only if

$$\bigcap_{\xi\in S^{d-1}}\Omega(\xi)=\{0\}.$$

This proposes that the Hardy--Littlewood--Sobolev inequality for the constrained space $\mathfrak{W}$ is characterized exactly by the cancellation condition that no nonzero vector belongs to every fiber of $\Omega$.