Toeplitz operators and Hankel operators on a Bergman space with an exponential weight on the unit ball (Q6583670)

Let \(A_{\psi}^{2}\) denote the Bergman space of all holomorphic functions on the ball \(\mathbf{B}_{n}\) square integrable with respect to the measure \(e^{-\psi}dV\), where \(\psi(z)=\frac{1}{1-|z|^{2}}\). Let also \(P_{\psi}\) be the orthogonal projection onto the Bergman space and let \(K_{\psi}\) be the corresponding Bergman kernel. For a finite Borel measure \(\mu\) on the ball \(\mathbf{B}_{n}\) one defines the operator\N\[\NT_{\mu}\colon A_{\psi}^{2}\rightarrow H(\mathbf{B}_{n}),\N\]\Nthe Toeplitz operator by the formula\N\[\NT_{\mu}f(z)=\int_{\mathbf{B}_{n}}K_{\psi}(z,w)f(w)e^{-\psi(w)}d\mu(w).\N\]\NThe authors characterize boundedness and compactness of the Toeplitz operators.\N\NTheorem 1.2 in the paper says that\N\begin{itemize}\N\item[(i)] \(T_{\mu}\) is bounded on \(A_{\psi}^{2}\) if and only if \(\mu\) is a Carleson measure for \(A_{\psi}^{2}\) if and only if \(\hat{\mu}_{r}\) is bounded on \(\mathbf{B}_{n}\).\N\item[(ii)] \(T_{\mu}\) is compact on \(A_{\psi}^{2}\) if and only if \(\mu\) is a vanishing Carleson measure for \(A_{\psi}^{2}\) if and only if \(\hat{\mu}_{r}(z)\rightarrow 0\) as \(|z|\rightarrow 1_{-}\).\N\end{itemize}\N\NA Borel measure on \(\mathbf{B}_{n}\) is a Carleson measure for \(A_{\psi}^{2}\) if there is a constant \(C>0\) such that\N\[\N\int_{\mathbf{B}_{n}}|f(z)|^{2}e^{-\psi(z)}d\mu(z)\leq C\int_{\mathbf{B}_{n}}|f(z)|^{2}e^{-\psi(z)}dV(z).\N\]\NThe measure \(\mu\) is called a vanishing Carleson measure if\N\[\N\lim_{j\rightarrow \infty}\int_{\mathbf{B}_{n}}|f_{j}(z)|^{2}e^{-\psi(z)}dV(z)=0\N\]\Nfor every bounded in \(A_{\psi}^{2}\) sequence \((f_{j})\) which converges to \(0\) uniformly on compact subsets.\N\NThe function \(\hat{\mu}_{r}\) is defined in the following way\N\[\N\hat{\mu}_{r}(z)=\frac{\mu(B_{\psi}(z,r))}{\text{Vol}(B_{\psi}(z,r))},\N\]\Nwhere \(B_{\psi}(z,r)\) the the ball of radius \(r\) centered at \(z\) in the Riemannian distance induced by the potential function \(\psi\).