We consider operators T satisfying a sparse domination property$$\begin{aligned} |\langle Tf,g\rangle |\le c\sum _{Q\in \mathscr {S}}\langle f\rangle _{p_0,Q}\langle g\rangle _{q_0',Q}|Q| \end{aligned}$$with averaging exponents \(1\le p_0<q_0\le \infty \). We prove weighted strong type boundedness for \(p_0<p<q_0\) and use new techniques to prove weighted weak type \((p_0,p_0)\) boundedness with quantitative mixed \(A_1\)–\(A_\infty \) estimates, generalizing results of Lerner, Ombrosi, and Pérez and Hytönen and Pérez. Even in the case \(p_0=1\) we improve upon their results as we do not make use of a Hörmander condition of the operator T. Moreover, we also establish a dual weak type \((q_0',q_0')\) estimate. In a last part, we give a result on the optimality of the weighted strong type bounds including those previously obtained by Bernicot, Frey, and Petermichl.

中文翻译：

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稀疏支配算子的弱和强类型 A 1 - A ∞ 估计。

我们认为算子T满足稀疏支配属性$$\begin{aligned} |\langle Tf,g\rangle |\le c\sum _{Q\in \mathscr {S}}\langle f\rangle _{p_0, Q}\langle g\rangle_{q_0',Q}|Q| \end{aligned}$$与平均指数\(1\le p_0<q_0\le \infty \)。我们证明了\(p_0<p<q_0\)的加权强类型有界性，并使用新技术证明加权弱类型\((p_0,p_0)\)有界性与定量混合\(A_1\) – \(A_\infty \ )估计，概括了 Lerner、Ombrosi 和 Pérez 以及 Hytönen 和 Pérez 的结果。即使在\(p_0=1\)的情况下我们改进了他们的结果，因为我们没有使用算子T的 Hörmander 条件。此外，我们还建立了对偶弱类型\((q_0',q_0')\)估计。在最后一部分中，我们给出了加权强类型边界的最优性，包括之前由 Bernicot、Frey 和 Petermichl 获得的结果。