We study weighted Bergman projections on the Hartogs triangle in C. We show that projections corresponding to exponentially vanishing weights have degenerate L mapping properties.

We prove that the Berezin transform is $L^p$ regular on a large class of domains in $\mathbb{C}^n$ and not $L^2$ Hartogs triangle.

The definition of classical holomorphic function spaces such as the Hardy space or Dirichlet on Hartogs triangle is not canonical. In this paper we introduce a natural family which includes some weighted Bergman spaces, candidate and space. For study L p mapping properties Szeg? projection respectively, whereas for prove it isometric to bidisc.

In this paper it is shown that the Hartogs triangle $${\mathbf{T}}$$ in $${\mathbf{C}}^2$$ a uniform domain. This implies Sobolev extension Furthermore, weak and strong maximal extensions of Cauchy-Riemann operator agree on triangle. These results have numerous applications. Among other things, they are used to study Dolbeault cohomology groups with coefficients complement .