Weak-type estimates for the Bergman projection on the polydisc and the Hartogs triangle. (English) Zbl 07276233

Huo, Zhenghui; Wick, Brett D.

Bull. Lond. Math. Soc. 52, No. 5, 891-906 (2020).

Summary: 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.

MSC:

32A25	Integral representations; canonical kernels (Szegő, Bergman, etc.)
32A36	Bergman spaces of functions in several complex variables

Keywords:

Bergman projection