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

Adam Osękowski
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Abstract

Let \({\mathcal {M}}_{\mathcal {D}}\) be the dyadic maximal operator on \({\mathbb {R}}^n\). The paper contains the identification of the best constant in the two-weight estimate

$$\begin{aligned} \Vert {\mathcal {M}}_{\mathcal {D}}f\Vert _{L^p(w)}\le C_{p,\sigma ,w}\Vert f\Vert _{L^p(\sigma ^{1-p})} \end{aligned}$$

under the assumption that the pair \((\sigma ,w)\) 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.

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凹凸条件下最大算子的尖锐两重估计值
让 \({\mathcal {M}}_{\mathcal {D}}\) 是 \({\mathbb {R}}^n\) 上的二元最大算子。本文包含对两重估计 $$\begin{aligned} 中最佳常数的识别。\Vert {\mathcal {M}}_{\mathcal {D}}f\Vert _{L^p(w)}\le C_{p,\sigma ,w}\Vert f\Vert _{L^p(\sigma ^{1-p})}\end{aligned}$$假设权重对((\sigma ,w)\)满足适当的碰撞条件。结果表明,在具有树状结构的抽象概率空间的更大范围内,该结果是正确的。
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