Hartogs Triangle Research Articles

Weighted Bergman projections on the Hartogs triangle

We prove the Lp regularity of the weighted Bergman projections on the Hartogs triangle, where the weights are powers of the distance to the singularity at the boundary. The restricted range of p is proved to be sharp. By using a two-weight inequality on the upper half plane with Muckenhoupt weights, we can consider a slightly wider class of weights.

Proper holomorphic mappings between generalized Hartogs triangles

Answering all questions—concerning proper holomorphic mappings between generalized Hartogs triangles—posed by Jarnicki and Plfug (First steps in several complex variables: Reinhardt domains, EMS Textbooks in Mathematics, European Mathematical Society Publishing House, 2008), we characterize the existence of proper holomorphic mappings between generalized Hartogs triangles and give their explicit form. In particular, we completely describe the group of holomorphic automorphisms of such domains and establish rigidity of proper holomorphic self-mappings on them. Moreover, we also complete the classification of proper holomorphic mappings in the class of complex ellipsoids which was initiated by Landucci and continued by Dini and Selvaggi Primicerio.

Bergman theory of certain generalized Hartogs triangles

The Bergman theory of domains $\{ |{z_{1} |^{\gamma}} < |{z_{2}} | < 1 \}$ in $\mathbb{C}^2$ is studied for certain values of $\gamma$, including all positive integers. For such $\gamma$, we obtain a closed form expression for the Bergman kernel, $\mathbb{B}_{\gamma}$. With these formulas, we make new observations relating to the Lu Qi-Keng problem and analyze the boundary behavior of $\mathbb{B}_{\gamma}(z,z)$.

On the holomorphic automorphism group of a generalized Hartogs triangle

In this paper, we completely determine the structure of the holomorphic automorphism group of a generalized Hartogs triangle and obtain natural generalizations of some results due to Landucci and Chen-Xu. These give affirmative answers to some open problems posed by Jarnicki and Pflug.

Correction: On the holomorphic automorphism group of a generalized Hartogs triangle

Correction: On the holomorphic automorphism group of a generalized Hartogs triangle

The Bergman projection on fat Hartogs triangles: $L^p$ boundedness

A class of pseudoconvex domains in $\mathbb{C}^{n}$ generalizing the Hartogs triangle is considered. The $L^p$ boundedness of the Bergman projection associated to these domains is established, for a restricted range of $p$ depending on the "fatness" of domains. This range of $p$ is shown to be sharp.

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

We prove optimal estimates for the mapping properties of the Bergman projection on the Hartogs triangle in weighted L spaces when p > 4 3 , where the weight is a power of the distance to the singular boundary point. For 1 < p ≤ 4 3 we show that no such weighted estimates are possible.

A class of domains with noncompact \overline{∂}-Neumann operator

The ∂ ¯ \overline {\partial } -Neumann operator (the inverse of the complex Laplacian) is shown to be noncompact on certain domains in complex Euclidean space. These domains are either higher-dimensional analogs of the Hartogs triangle or have such a generalized Hartogs triangle imbedded appropriately in them.

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

The regularity of the \(\overline{\partial }\)-problem on the domain \(\{\left|{z_1}\right|\!<\!\left|{z_2}\right|\!<\!1\}\) in \(\mathbb C ^2\) 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.

The classification of proper holomorphic mappings between special Hartogs triangles of different dimensions

The authors discuss the proper holomorphic mappings between special Hartogs triangles of different dimensions and obtain a corresponding classification theorem.

Rigidity of Proper Self-Mapping on Some Kinds of Generalized Hartogs Triangle

It is proved that every proper holomorphic self-mapping of some kinds of Generalized Hartogs Triangles is an automorphism, and its explicit expression is given.

Jensen measures, hyperconvexity and boundary behaviour of the pluricomplex Green function

Let be a bounded domain in CN. Let z be a point in and let Jz be the set of all Jensen measures on with barycenter at z with respect to the space of functions continuous on and plurisubharmonic in . The authors prove that is hyperconvex if and only if, for every z 2 @ , measures in Jz are supported by @ . From this they deduce that a pluricomplex Green function g(z,w) with its pole at w continuously extends to @ with zero boundary values if and only if is hyperconvex. Then the authors give a criterion for Reinhardt domains to be hyperconvex and explicitly compute the pluricomplex Green function on the Hartogs triangle. The last sections are devoted to the boundary behaviour of pluricomplex Green functions. Such a function has Property (P0) at a point w0 2 @ if limw!w0 g(z,w) = 0 for every z 2 . If the convergence is uniform in z on compact subsets of r{w0}, then w0 has Property (P0). Several sufficient conditions for points on the boundary with these properties are given. (Less)