The \(L^{p}\) boundedness of the Bergman projection for a class of bounded Hartogs domains. (English) Zbl 1361.32005
The author studies the Bergman projection on a class of bounded Hartogs domains that generalize the case of the Hartogs triangle \(\mathbb{H}=\{(z_1,z_2)\in\mathbb{C}^2: \left| z_1\right|<\left| z_2\right|<1\}\). In an analogous fashion as to what is known for the Hartogs triangle, the main result of the paper shows that, for the class of domains considered, the Bergman projection is bounded on \(L^p\) if and only if \(\frac{2n}{n+1}<p<\frac{2n}{n-1}\)
The exact class of domains studied is defined in the following way. For \(j=1,\ldots, \ell\) let \(\Omega_j\) be a domain in \(\mathbb{C}^{k_j}\) and let \(\phi_j:\Omega_j\to \mathbb{B}^{k_j}\) be a biholomorphic map, where \(\mathbb{B}^k\) is the unit ball in \(\mathbb{C}^k\). For the sequence \(m_0=0\), \(m_j=\sum_{s=1}^{j} k_s\) and \(m_l=k<n\), let \(\tilde{z}_j=(z_{m_{j-1}+1},\ldots, z_{m_j})\). The domains considered are then defined by \[ \mathbb{H}^n_{k_j, \phi_j}=\Big\{z\in\mathbb{C}^n: \max_{1\leq j\leq l} \left| \phi_j(\tilde{z}_j)\right|<\left| z_{k+1}\right|<\cdots<| z_{n}|<1\Big\}. \] The method of proof is an application of Schur’s Test.
The exact class of domains studied is defined in the following way. For \(j=1,\ldots, \ell\) let \(\Omega_j\) be a domain in \(\mathbb{C}^{k_j}\) and let \(\phi_j:\Omega_j\to \mathbb{B}^{k_j}\) be a biholomorphic map, where \(\mathbb{B}^k\) is the unit ball in \(\mathbb{C}^k\). For the sequence \(m_0=0\), \(m_j=\sum_{s=1}^{j} k_s\) and \(m_l=k<n\), let \(\tilde{z}_j=(z_{m_{j-1}+1},\ldots, z_{m_j})\). The domains considered are then defined by \[ \mathbb{H}^n_{k_j, \phi_j}=\Big\{z\in\mathbb{C}^n: \max_{1\leq j\leq l} \left| \phi_j(\tilde{z}_j)\right|<\left| z_{k+1}\right|<\cdots<| z_{n}|<1\Big\}. \] The method of proof is an application of Schur’s Test.
Reviewer: Brett Wick (St. Louis)
MSC:
| 32A25 | Integral representations; canonical kernels (Szegő, Bergman, etc.) |
| 32A36 | Bergman spaces of functions in several complex variables |
| 32A07 | Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010) |
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