Sawyer-type inequality for the Hardy–Littlewood maximal operator with rearrangement weights

Fix exponents $q3e=1$ and $\varsigma>0$ (or $03cvarsigma3c=1$), and write

$$\varrho=1+\varsigma(q-1).$$

Let $u$ and $v$ be positive, measurable functions such that $u^{\varsigma}\in A_{\varrho}^{\mathcal R}$ and $u^{\varsigma}v^{\varrho}\in A_r^{\mathcal R}$ for some $r3e=1$. Sawyer-type conjecture. There exists a function $\phi:[1,\infty)^2\longrightarrow[0,\infty)$, increasing in each variable, such that for every measurable function $f$,

$$\left\Vert \frac{M(|f|^{q/\varrho}u^{\frac{1-\varsigma}{\varrho}})}{v} \right\Vert_{L^{\varrho,\infty}(u^{\varsigma}v^{\varrho})}^{\varrho}\leq \phi([u^{\varsigma}]_{A_{\varrho}^{\mathcal R}},[u^{\varsigma}v^{\varrho}]_{A_r^{\mathcal R}})\Vert f\Vert_{L^{q,1}(u)}^q.$$

This inequality is conjectured because it would extend the preceding extrapolation theorem to arbitrary exponents $1\leq q<p$; its resolution is not supplied in the paper.