Weak-type weighted inequality characterization for the maximal operator

Let $X$ be a space of homogeneous type, let $M$ be the uncentered Hardy–Littlewood maximal operator, and let $p(39)\in\mathrm{LH}$ satisfy $p_+<\infty$. Let $w$ be a weight, and let $A_{p(39)}$ be the class of weights such that there is a constant $K$ with

$$\|w\chi_B\|_{p(39)}\|w^{-1}\chi_B\|_{p'(39)}\leq K\mu(B)$$

for every ball $B$. Weak-type weighted maximal-operator conjecture. The inequality

$$\|t\chi_{\{x\in X: Mf(x)>t\}}w\|_{p(39)}\leq C\|fw\|_{p(39)}$$

for the relevant functions $f$ and levels $t>0$ holds if and only if $w\in A_{p(39)}$. This is the variable-exponent analogue of the classical weak- and strong-type weighted theory for the Hardy–Littlewood maximal operator; the stated text identifies it as an open problem, while the corresponding strong-type characterization is given as the main theorem.