Carleson measures on the generalized Hartogs triangles
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 \(\{\psi(a_j)^{\frac12(1-\alpha)p-1}\widehat{\mu^p}_r(a_j)\}_{j=1}^\infty\) is bounded. Theorem. If \(\mu\) is a vanishing \(p\)-Carleson of \(A^2_\alpha(H)\), then for any \(r\in(0,1)\) we have \(\psi(z)^{\frac12(1-\alpha)p-1}\widehat{\mu^p}_r(z)\longrightarrow0\) as \(z\longrightarrow\partial H\setminus\{0\}\).
scientific article; zbMATH DE number 7474369
scientific article