A sharp two-weight estimate for the maximal operator under a bump condition

Let \({\mathcal {M}}_{\mathcal {D}}\) ${\mathcal{M}}_{\mathcal{D}}$ be the dyadic maximal operator on \({\mathbb {R}}^n\) ${\mathbb{R}}^n$ . The paper contains the identification of the best constant in the two-weight estimate \(\begin{aligned} \Vert {\mathcal {M}}_{\mathcal {D}}f\Vert _{L^p(w)}\le C_{p,\sigma ,w}\Vert f\Vert _{L^p(\sigma ^{1-p})} \end{aligned}\) $$\begin{array}{c}\hfill \Vert {\mathcal{M}}_{\mathcal{D}}{f\Vert }_{L^p(w)}\le C_{p,\sigma ,w}{\Vert f\Vert }_{L^p({\sigma }^{1-p})}\end{array}$$ under the assumption that the pair \((\sigma ,w)\) $(\sigma ,w)$ of weights satisfies an appropriate bump condition. The result is shown to be true in the larger context of abstract probability spaces equipped with a tree-like structure.