Characterize the endpoint class for \(L_w^{p,1}\to L_w^{q,\infty}\) boundedness

Establish whether the condition \(w\in A_{(p,1),(q,\infty)}\) is sufficient for boundedness of the fractional maximal operator \(M_\alpha:L_w^{p,1}\to L_w^{q,\infty}\) when \(p>1\).

Background

The paper proves that $A_{(p,1),(q,\infty)}$ is necessary for the stated multiplier-weighted weak-type estimate. It also proves sufficiency under the stronger condition $w\in A_{(p,p),(q,\infty)}$, and explicitly notes that the latter class is properly contained in the necessary class. Thus the exact characterization remains unresolved.