Carleson measures and reproducing kernel thesis in Dirichlet-type spaces (Q2856435)

For a positive Borel measure \(\mu\) on the boundary \(\mathbb T\) of the unit disc \(\mathbb D\) in \(\mathbb C\), let \(P_\mu(z)\) denote the Poisson integral of \(\mu\). The corresponding Dirichlet space \(\mathcal D(u)\) is defined to be the space of holomorphic functions \(f\) on \(\mathbb D\) for which NEWLINE\[NEWLINE \mathcal D_\mu(f) = \int_{\mathbb D} |f^\prime(z)|^2 P_\mu(z)\, dA(z) <\infty NEWLINE\]NEWLINE with norm \(\|f\|_\mu^2 = \|f\|_2^2 + \mathcal D_\mu(f) \), where \(\|f\|_2\) is the Hardy space \(H^2\) norm of \(f\). Taking \(\mu\) as the normalized Lebesgue measure on \(\mathbb T\) gives the classical Dirichlet space \(\mathcal D\). In [Mich. Math. J. 38, No. 3, 355--379 (1991; Zbl 0768.30040)], \textit{S. Richter} and \textit{C. Sundberg} proved that \(f\in\mathcal D(\delta_\lambda)\), where \(\delta_\lambda\) is the point mass at \(\lambda\in\mathbb T\), if and only if \(f= c + (z-\lambda)g\), where \(c\) is the nontangential limit of \(f\) at \(\lambda\), denoted by \(f(\lambda)\), and \(g\in H^2\) with \(\|g\|_2^2 = \mathcal D_{\delta_\lambda}(f)\), i.e., NEWLINE\[NEWLINE \mathcal D_{\delta_\lambda}(f) = \left\|\frac{f-f(\lambda)}{z-\lambda}\right\|_2^2. NEWLINE\]NEWLINE In Section 3 of the paper the authors extend the Richter and Sundberg result to the case where \(\mu = \sum_{j=1}^n \alpha_j \delta_{\lambda_j}\), \(\alpha_j >0\), \(\lambda_j\in\mathbb T\). For such a \(\mu\), \(f\in\mathcal D(\mu)\) if and only if there exists a unique \(g\in H^2\) and a unique polynomial \(p\), \(\text{ deg\,}p \leq n-1\), such that NEWLINE\[NEWLINE f = p + \prod_{j=1}^n (z-\lambda_j) g NEWLINE\]NEWLINE with \(\|g\|_2 \leq C\|f\|_\mu\). Using this result the authors prove that a finite positive measure \(\nu\) on \(\mathbb D\) is a Carleson measure for \(\mathcal D(\mu)\) if and only if \(\prod_{j=1}^n |z-\lambda_j|^2d\nu(z)\) is a Carleson measure for \(H^2\). In Sections 5 and 6 the authors provide a characterization of Carleson measures for \(\mathcal D(\mu)\), \(\mu\) as above, in terms of the reproducing kernel \(k^\mu\) for \(\mathcal D(\mu)\).