Hankel measures for Hardy spaces
In the paper the authors study Hankel measures on the unit disk \(\mathbb{D}\). A complex Borel measure \(\mu\) on \(\mathbb{D}\) is called a Hankel measure if there exists a constant \(C>0\) such that \[ \Bigl{|}\int_{\mathbb{D}}f^{2}(z)d\mu(z)\Bigr{|}\leq C\|f\|_{H^{2}}^{2}, \] where \(\|\cdot\|_{H^{p}}\) is the Hardy space \(H^{p}\) norm. It is proved (Theorem 1) that \(\mu\) is a Hankel measure if \[ \Bigl{|}\int_{\mathbb{D}}f(z)d\mu(z)\Bigr{|}\leq C\|f\|_{H^{1}}. \] This in turn is equivalent to the fact that \(P_{1}(\bar{\mu})\in BMOA\), where the integral transform \(P_{1}\) is defined in the following way \[ P_{1}(\mu)(w)=\int_{\mathbb{D}}\frac{1}{1-w\bar{z}}d\mu(z). \] The main result of the paper is an answer to the question raised by J. Xiao in [\textit{J. Xiao}, Bull. Austral. Math. Soc. 62, 135--140 (2000; Zbl. 0977.30019)]. Theorem 4 says that there is a measure \(\mu\) on \(\mathbb{D}\) which is not a Hankel measure but \[ \sup_{w\in \mathbb{D}}|P_{2}(\bar{\mu})(w)|=\sup_{w\in \mathbb{D}}\Bigl{|}\int_{\mathbb{D}}\frac{1-|w|^{2}}{(1-\bar{w}z)^{2}}d\mu(z)\Bigr{|}<\infty. \] However if \(\mu\) is a positive Borel measure supported on the real interval \([0,1)\) the the above condition is sufficient for \(\mu\) to be a Hankel measure (Theorem 9). This is also equivalent to \(P_{1}(\mu)\) belonging to (some) all \(\mathcal{Q}_{p}\) spaces \(0<p<\infty\). Lastly, it is proved (Theorem 10) that a finite real-valued Borel measure on \(\mathbb{D}\) is a Borel measure if and only if the balayage of \(\mu\) \[ g(e^{i\theta})=\int_{\mathbb{D}}\frac{1-|z|^{2}}{|1-ze^{-i\theta}|^{2}}d\mu(z) \] is in BMO.
scientific article; zbMATH DE number 7343440
scientific article