发表机构
Universidad Autónoma de Madrid; Università degli studi di Milano-Bicocca; Universidad de Málaga（马德里自治大学; 米兰比可卡大学; 马拉加大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文建立适应嵌套族的稀疏支配框架，证明Carleson条件与广义稀疏条件等价，并应用于树上的Bergman投影算子，获得新的加权不等式与有界性结果。
AI 中文摘要
我们发展了一个适应于一般测度空间中嵌套族（可能包含点质量）的稀疏支配框架。我们证明了Carleson条件等价于通常稀疏条件的轻微推广，并利用这一刻画获得了相关平均算子的一权和二权估计，以及相应的逆检验条件。我们将一般结果应用于局部有限树上的积分算子。树自然配备两种嵌套几何：扇区和修剪扇区。我们证明了调和Bergman投影算子的自然版本被相应的平均算子逐点稀疏支配，这取决于树上的测度是否满足倍测度条件，并表明在一般情况下这种区分是必要的。对于正Bergman投影算子，我们获得了逐点可比性。作为推论，我们获得了Bergman投影算子及一类Toeplitz型乘子的新的加权不等式和有界性结果。
英文摘要
We develop a sparse domination framework adapted to nested families in general measure spaces, possibly including point masses. We show that the Carleson condition is equivalent to a slight generalization of the usual sparse condition, and use this characterization to obtain one- and two-weight estimates for the associated averaging operators, together with converse testing conditions. We apply our general results to integral operators on locally finite trees. Trees come naturally equipped with two nested geometries: sectors and pruned sectors. We prove pointwise sparse domination of natural versions of the harmonic Bergman projector by the corresponding averaging operators, depending on whether the measure on the tree is doubling or not, and show that the distinction is necessary in general. For the positive Bergman projector, we obtain pointwise comparability. As a consequence, we obtain new weighted inequalities and boundedness results for the Bergman projector and a class of Toeplitz-like multipliers.
