Weak-type weighted Sobolev conjecture for the nonlinear operator N

Let $n'=n/(n-1)$, let $w$ belong to the weight class $A_{1,n'}$, and let $N$ be the nonlinear operator considered above. The notation $L^{n',\infty}(w^{n'})$ denotes the weighted weak Lebesgue space, and $[w]_{A_{1,n'}}$ is the corresponding weight characteristic.

Weak-type weighted Sobolev conjecture. For such $w$ and $f$,

$$\|Nf\|_{L^{n',\infty}(w^{n'})} \leq c\, [w]_{A_{1,n'}}^{2+\frac1{n'}}\, \|w\nabla f\|_{L^1(\mathbb R^n)}.$$

This conjecture is motivated by the preceding weighted Sobolev estimate and seeks an endpoint bound without the blow-up in the exponent as $p\to1$. Its resolution is not supplied in the given text.