Sobolev regularity of the $$\overline{\partial }$$ -equation on the Hartogs triangle

Abstract

The regularity of the ∂ -problem on the domain {|z 1 | < |z 2 | < 1} in C 2 is studied using L 2 -methods. Estimates are obtained for the canonical solution in weighted L 2 -Sobolev spaces with a weight that is singular at the point (0, 0). In particular, the singularity of the Bergman projection for the Hartogs triangle is contained at the singular point and it does not propagate.

$L^p$ mapping properties of the Bergman projection on the Hartogs triangle

“…Consequently, one can obtain estimates which take into account the behavior of functions and forms near the singular points. For the Hartogs triangle, this was done in [CS13], where estimates for the canonical solution of the ∂-problem were obtained in weighted Sobolev spaces. While the Hartogs triangle has a non-Lipschitz boundary, as a complex manifold it has a very simple structure: it is biholomorphic to the product D * × D of the punctured unit disc D * = { z ∈ C | 0 < |z| < 1 } and the unit disc D. Consequently, one can pull back problems on H to problems on D * × D, and one gets weights coming from the Jacobian factor.…”

“…While the Hartogs triangle has a non-Lipschitz boundary, as a complex manifold it has a very simple structure: it is biholomorphic to the product D * × D of the punctured unit disc D * = { z ∈ C | 0 < |z| < 1 } and the unit disc D. Consequently, one can pull back problems on H to problems on D * × D, and one gets weights coming from the Jacobian factor. This technique was used in [MM92,CS13] to study function theory on H. Here we use the same method to study the mapping properties of B H in L p spaces. This paper is organized as follows.…”

Bergman–Toeplitz operators on fat Hartogs triangles

“…The model of Hartogs triangles and their variants has recently attracted particular attention through the study of several problems in complex analysis; see e.g. [9,11,6,5,13]. The fat Hartogs triangle Ω k is a pseudoconvex domain but not a hyperconvex domain (see e.g.…”

Weighted Bergman projections on the Hartogs triangle

“…In [CS11] and [CS13], Chakrabarti and Shaw focus on the ∂-equation and the corresponding Sobolev regularity on the product domains and the Hartogs triangle. In [Che13], the author shows the (ordinary) Bergman projection is L p bounded on the Hartogs triangle if and only if p ∈ 4 3 , 4 .…”