In this paper, we investigate the weak‐type regularity of the Bergman projection. The two domains we focus on are the polydisc and the Hartogs triangle. For the polydisc, we provide a proof that the weak‐type behavior is of ‘ $L\log L$’ type. This result is likely known to the experts, but does not appear to be in the literature. For the Hartogs triangle, we show that the operator is of weak‐type (4,4); settling the question of the behavior of the projection at this endpoint. At the other endpoint of interest, we show that the Bergman projection is not of weak‐type $(\frac{4}{3},\frac{4}{3})$ and provide evidence as to what the correct behavior at this endpoint might be.

中文翻译：

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多盘和Hartogs三角形上Bergman投影的弱型估计

在本文中，我们研究了伯格曼投影的弱型正则性。我们关注的两个领域是多碟和Hartogs三角形。对于多碟，我们提供证明弱类型行为为“$\mathrm{大号}\mathrm{日志}\mathrm{大号}$'类型。专家可能已经知道了这个结果，但是在文献中似乎没有。对于Hartogs三角形，我们证明算子是弱类型（4,4）；解决这个端点的投影行为问题。在另一个有趣的终点，我们证明了伯格曼投影不是弱类型$（\frac{4}{3}，\frac{4}{3}）$ 并提供有关此端点可能发生的正确行为的证据。