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Weighted Bergman Projection on the Hartogs Triangle  [PDF]
Liwei Chen
Mathematics , 2014,
Abstract: We prove the $L^p$ regularity of the weighted Bergman projection 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.
Sobolev regularity of the $\bar{\partial}$-equation on the Hartogs triangle  [PDF]
Debraj Chakrabarti,Mei-Chi Shaw
Mathematics , 2011,
Abstract: The regularity of the $\bar{\partial}$-problem on the domain $\{|{z_1}|<|{z_2}|<1\}$ 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 $(0,0)$. The canonical solution for $\dbar$ with weights is exact regular in the weighted Sobolev spaces away from the singularity $(0,0)$. 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  [PDF]
Debraj Chakrabarti,Yunus E. Zeytuncu
Mathematics , 2014,
Abstract: We prove optimal estimates for the mapping properties of the Bergman projection on the Hartogs triangle in weighted $L^p$ spaces when $p>\frac{4}{3}$, where the weight is a power of the distance to the singular boundary point. For $1
The $L^p$ boundedness of the Bergman projection for a class of bounded Hartogs domains  [PDF]
Liwei Chen
Mathematics , 2013,
Abstract: We generalize the Hartogs triangle to a class of bounded Hartogs domains, and we prove that the corresponding Bergman projections are bounded on $L^p$ if and only if $p$ is in the range $(\frac{2n}{n+1},\frac{2n}{n-1})$.
The Bergman projection on fat Hartogs triangles: L^p boundedness  [PDF]
L. D. Edholm,J. D. McNeal
Mathematics , 2015,
Abstract: 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.
Regularity of Canonical Operators and Nebenhülle: Hartogs Domains  [PDF]
Yunus E. Zeytuncu
Mathematics , 2013,
Abstract: We relate the regularity of the Bergman projection operator and the canonical solution operator to the Nebenh\"ulle of complete Hartogs domains.
The Bergman kernels on Cartan-Hartogs domain
Weiping Yin
Chinese Science Bulletin , 1999, DOI: 10.1007/BF02887114
Abstract: The main point is the calculation of the Bergman kernel for the so-called Cartan-Hartogs domains. The Bergman kernels on four types of Cartan-Hartogs domains are given in explicit formulas. First by introducing the idea of semi-Reinhardt domain is given, of which the Cartan-Hartogs domains are a special case. Following the ideas developed in the classic monograph of Hua, the Bergman kernel for these domains is calculated. Along this way, the method of “inflation”, is made use of due to Boas, Fu and Straube.
The Bergman Kernel on some Hartogs domains  [PDF]
Zhenghui Huo
Mathematics , 2015,
Abstract: We obtain new explicit formulas for the Bergman kernel function on two families of Hartogs domains. To do so, we first compute the Bergman kernels on the slices of these Hartogs domains with some coordinates fixed, evaluate these kernel functions at certain points off the diagonal, and then apply a first order differential operator to them. We find, for example, explicit formulas for the kernel function on $$\{(z_1,z_2,w)\in\mathbb C^3:e^{|w|^2}|z_1|^2+|z_2|^2<1\}$$ and on $$\{(z_1,z_2,w)\in\mathbb C^3:|z_1|^2+|z_2|^2+|w|^2<1+|z_2w|^2\;{\rm and} \;|w|<1\}.$$ We use our formulas to determine the boundary behavior of the kernel function of these domains on the diagonal.
Bergman theory on fat Hartogs triangles  [PDF]
Luke Edholm
Mathematics , 2015,
Abstract: The Bergman theory of domains $\{ |{z_{1} |^{k}} < |{z_{2}} | < 1 \}$ in $\mathbb{C}^2$ is studied for certain values of $k$, including all positive integers. For these $k$, we obtain a closed form expression for the Bergman kernel, $\mathbb{B}_{k}$. Using these formulas, we make new discoveries about the Lu Qi-Keng problem and are able to explicitly analyze the behavior of $\mathbb{B}_{k}(z,z)$ as $z$ tends to singular boundary points.
A remark on the dimension of the Bergman space of some Hartogs domains  [PDF]
Piotr Jucha
Mathematics , 2009, DOI: 10.1007/s12220-010-9182-8
Abstract: Let D be a Hartogs domain of the form D={(z,w) \in CxC^N : |w| < e^{-u(z)}} where u is a subharmonic function on C. We prove that the Bergman space of holomorphic and square integrable functions on D is either trivial or infinite dimensional.
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