Separated bump conjecture for strong-type inequalities of Riesz potentials

Let $0<\alpha<n$ and let $p,q$ satisfy $1<p\leq q<\infty$. For weights $(u,\sigma)$, write $p'=p/(p-1)$ and let $\Phi$ and $\Psi$ be Young functions, with complementary functions $\bar{\Phi}$ and $\bar{\Psi}$. Let $\|\cdot\|_{\Phi,Q}$ denote the localized Orlicz norm on a cube $Q$, and let $B_r$ denote the corresponding class of Young functions. The pair $(u,\sigma)$ satisfies the separated bump conditions if

$$\sup_Q |Q|^{\frac{\alpha}{n}+\frac1q-\frac1p}\|u^{\frac1q}\|_{\Phi,Q} \|\sigma^{\frac{1}{p'}}\|_{p',Q} <\infty,$$

and

$$\sup_Q |Q|^{\frac{\alpha}{n}+\frac1q-\frac1p} \|u^{\frac{1}{q}}\|_{q,Q} \|\sigma^{\frac{1}{p'}}\|_{\Psi,Q}<\infty.$$

Separated bump conjecture. Given $0<\alpha<n$ and $1<p\leq q<\infty$, the strong-type inequality for the Riesz potential holds for every pair of weights $(u,\sigma)$ satisfying these conditions, whenever $\bar{\Phi}\in B_{q'}$ and $\bar{\Psi}\in B_p$.

This conjecture proposes that the single two-weight bump condition can be replaced by two conditions, each involving only one bumped factor. A related result is proved in the paper for nearby ranges and for log and loglog bumps, but the full separated-bump assertion remains open in the source.