Carleson measures on the generalized Hartogs triangles
type: Item
instance of: scholarly article
title: Carleson measures on the generalized Hartogs triangles
zbMATH Open document ID: 1487.32029
author: Shuo Zhang
published in: Journal of Mathematical Analysis and Applications
publication date: 2022-02-16T00:00:00Z
review text: The author characterizes Carleson measures and vanishing Carleson measures of weighted Bergman spaces on the generalized Hartogs triangle \[H=H^n_{\{k_i\}, \gamma}:=\{z\in\mathbb C^n: \max\limits_{i=1,\dots,\ell}\|\widetilde z_i\|<|z_{k+1}|^\gamma<\dots<|z_n|^\gamma<1\},\] where \(n\geq2\), \(\gamma, k_1,\dots,k_\ell\in\mathbb N\), \(k:=\sum_{i=1}^\ell k_i<n\), \(\widetilde z_i:=(z_{m_{i-1}+1},\dots,z_{m_i})\), \(m_0:=0\), \(m_i:=\sum_{j=1}^i k_j\). Define \(\varphi(z):=|z_{k+1}|^{-2k\gamma}\prod_{j=k+2}^n|z_j|^{-2}\) and \[\psi(z):=\prod_{i=1}^\ell(1-\|\frac{\widetilde z_i}{z_{k+1}^\gamma}\|^2)^{-(k_i+1)}\cdot\prod_{j=k+1}^n(1-|\frac{z_j}{z_{j+1}}|^2)^{-2},\] \(z\in H\). Let \(\mu\) be a finite positive Borel measure on \(H\) and let \(1\leq p<+\infty\). We say that \(\mu\) is a \(p\)-Carleson measure (resp. vanishing \(p\)-Carleson measure) of the weighted Bergman space \(A^2_\alpha(H):=\{f\in\mathcal O(H): \int_H|f|^2\psi^\alpha dV<+\infty\}\) if the inclusion operator \(A^2_\alpha(H)\longrightarrow L^p(H,\mu)\) is bounded (resp. compact). Let \(\mu\) be a finite positive Borel measure on \(H\), \(p, \alpha\in\mathbb R\), \(2\leq p\leq\infty\), \(-1<\alpha<\min\limits_{i=1,\dots,\ell}\frac1{k_i+1}\). Let \(\mu^p\) be the measure on \(H\) defined by \(d\mu^p=\varphi^{p/2}d\mu\). The main results of the paper are the following two theorems: Theorem. The following conditions are equivalent: (1) \(\mu\) is a \(p\)-Carleson measure of \(A^2_\alpha(H)\); (2) the function \(H\ni z\longmapsto\psi(z)^{-\frac12p\alpha}B^p\mu(z)\) is bounded, where \(B^p_\mu(z):=\int_H|\frac{K_H(z,w)}{K_H(z,z)^{1/2}}|^pd\mu(w)\); (3) the function \(H\ni z\longmapsto\psi(z)^{\frac12(1-\alpha)p-1}\widehat{\mu^p}_r(z)\) is bounded, where \(\widehat{\mu^p}_r(z):=\frac{\mu^p(B^c_H(z,r))}{V(B^c_H(z,r))}\), \(B^c_H(z,r):=\{w\in H: \tanh c_H(w,z)<r\}\), \(c_H\) stands for the Möbius distance; (4) for any \(r\in(0,1)\) and \(r\)-lattice \(\{a_j\}_{j=1}^\infty\) the sequence \(\{\psi(a_j)^{\frac12(1-…
reviewed by: Marek Jarnicki
Mathematics Subject Classification ID: 32A36
Mathematics Subject Classification ID: 32Q02
Mathematics Subject Classification ID: 32F45
zbMATH DE Number: 7474369
zbMATH Keywords: Carleson measure
zbMATH Keywords: generalized Hartogs triangle
zbMATH Keywords: weighted Bergman space
MaRDI profile type: Publication
full work available at URL: j.jmaa.2022.126027
OpenAlex ID: W4206978435
cites work: Carleson measures and uniformly discrete sequences in strongly pseudoconvex domains
cites work: Toeplitz operators and Carleson measures in strongly pseudoconvex domains
cites work: Skew Carleson measures in strongly pseudoconvex domains
cites work: Toeplitz operators and skew Carleson measures for weighted Bergman spaces on strongly pseudoconvex domains
cites work: A Lower Bound for the Bergman Kernel and the Bourgain-Milman Inequality
cites work: The \(L^{p}\) boundedness of the Bergman projection for a class of bounded Hartogs domains
cites work: Q4865559
cites work: Q3965067
cites work: The pseudohyperbolic metric and Bergman spaces in the ball
cites work: Bergman subspaces and subkernels: degenerate \(L^p\) mapping and zeroes
cites work: On the product property of the pluricomplex Green function
cites work: The Bergman metric on hyperconvex domains
cites work: The Carathéodory pseudodistance has the product property
cites work: Invariant distances and metrics in complex analysis
cites work: A Carleson Measure Theorem for Bergman Spaces
cites work: Carleson measures and balayage for Bergman spaces of strongly pseudoconvex domains
cites work: Carleson measures for Besov spaces on the ball with applications
cites work: Carleson measures and Toeplitz operators on doubling Fock spaces
cites work: Q3958740
cites work: Special Toeplitz operators on a class of bounded Hartogs domains
cites work: Carleson measures on the weighted Bergman spaces with Békollé weights
cites work: Lp boundedness for the Bergman projections over n-dimensional generalized Hartogs triangles
cites work: Spaces of Holomorphic Functions in the Unit Ball
DOI: 10.1016/J.JMAA.2022.126027
description: scientific article; zbMATH DE number 7474369