Carleson measures for Hilbert spaces of analytic functions on the complex half-plane (Q323826)

Let \(\mathcal{H}\) denote a Hilbert space of analytic functions on some domain \(\Omega\) in the complex plane \(\mathbb{C}\), and let \(\mu\) be a positive Borel measure on \(\Omega\). The measure \(\mu\) is said to be a \textit{Carleson measure} if there exists a constant \(C\) depending only on \(\mu\) such that for every function \(f\in \mathcal{H}\) we have NEWLINE\[NEWLINE\int_{\Omega}|f(z)|^2 d\mu (z)\leq C\|f\|^2_{\mathcal{ H}}.NEWLINE\]NEWLINE This notion was first introduced by Lenart Carleson in his proof of the the so-called Corona Theorem for the space of bounded analytic functions on the unit disk, \(H^\infty(\mathbb{D})\). Around 1960, Carleson provided a complete characterisation of Carleson measures for the Hardy space \(H^p(\mathbb{D})\) where \(1\leq p<\infty\). He proved that a necessary and sufficient condition for \(\mu\) to be a Carleson measure is that \(\mu (S)\leq Ch\) for some constant \(C\) and all \textit{Carleson squares} NEWLINE\[NEWLINES=\{ re^{i\theta}: 1-h\leq r<1, |\theta -\theta_0|\leq h\}NEWLINE\]NEWLINE of side-length \(h\) at the boundary of the disk. It is a result of W. W. Hastings that the Carleson measures for the Bergman spaces have the analogous description \(\mu (S)\leq Ch^2\). Then appeared the description of Carleson measures for the Bergman space by D. H. Luecking in terms of pseudohyperbolic metric, as well as the characterisation of Carleson measures for the Dirichlet space given by D. Stegenga.NEWLINENEWLINEIn the paper under review, the author replaces the unit disk by the right half plane NEWLINE\[NEWLINE\mathbb{C}_+:=\{z=x+iy \in \mathbb{C}: x>0\},NEWLINE\]NEWLINE and gives a necessary and sufficient condition for \(\mu\) to be a Carleson measure for the Dirichlet space on the right half plane \(\mathcal D(\mathbb{C}_+)\). This is the space of analytic functions \(f\) on \(\mathbb{C}_+\) equipped with the norm NEWLINE\[NEWLINE\|f\|^2_{\mathcal D(\mathbb{C}_+)}:=\|f\|^2_{\mathcal H^2(\mathbb{C}_+)}+\|f^\prime\|^2_{\mathcal B^2(\mathbb{C}_+)}NEWLINE\]NEWLINE where NEWLINE\[NEWLINE\|f\|^2_{\mathcal H^2(\mathbb{C}_+)}=\sup_{x>0}\int_{-\infty}^\infty|f(x+iy)|^2 \frac{dy}{2\pi}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\|f^\prime\|^2_{\mathcal{B}^2(\mathbb{C}_+)}=\int_{\mathbb{C}_+}|f^\prime (z)|^2\, \frac{dx\, dy}{\pi}.NEWLINE\]NEWLINE We recall that \(\mathcal {H}^2(\mathbb{C}_+)\) denotes the Hardy space on \(\mathbb{C}_+\), and \(\mathcal{B}^2_{(\mathbb{C}_+)}\) stands for the Bergman space on the right half plane.NEWLINENEWLINE Among other things, the author proves that a positive Borel measure \(\mu\) on \(\mathbb{C}_+\) is a Carleson measure for the Dirichlet space on \(\mathbb{C}_+\) if and only if there exists a constant \(C\) such that for every \(G\in L^2(\mathbb{C}_+, \mu)\) we have NEWLINE\[NEWLINE\int_{\mathbb{C}_+}\left |\int_{\mathbb{C}_+}\frac{G(\zeta)}{z+\overline{\zeta}}d\mu(\zeta)\right |^2\frac{dx\, dy}{\pi e^{2\,Re(z)}}\leq C\int_{\mathbb{C}_+}|G|^2d\mu.NEWLINE\]NEWLINE The characterisation of Carleson measures for the spaces \(\mathcal {D}^\alpha(\mathbb{C}_+)\) is also discussed. The last section of the paper is devoted to Carleson embeddings for trees.