Sampling measures for Bergman spaces on the unit disk
For the open unit disc \(\mathbb{D}\) in the complex plane and \(\alpha>-1\), define \(dA_\alpha(z)= (1-|z|^2)^\alpha dA\), where \(dA\) is integration with respect to Lebesgue area measure on \(\mathbb{D}\). The weighted Bergman space \(A^p(\alpha)\) is defined to be the collection of all analytic functions \(f\) such that \(\|f\|_{p,\alpha}= [\int_{\mathbb{D}}|f|^p dA_\alpha]^{1/p}< \infty\). A measure \(\tau\) is called a Carleson measure for \(A^p(\alpha)\) if there exists a constant \(c_1>0\) such that \(\int_{\mathbb{D}}|f|^p d\tau\leq c_1\|f\|^p_{p,\alpha}\) for all \(f\in A^p(\alpha)\). Let \(C\) denote the set of all measures \(\mu\) such that \(\tau= (1-|z|^2)^{2+\alpha}\mu\) is a Carleson measure for \(A^p(\alpha)\). Finally, a measure \(\mu\) will be called a sampling measure for \(A^p(\alpha)\) if there is a constant \(\Lambda> 0\) such that \[ {1\over\Lambda}\|f\|^p_{p,\alpha}\leq \int_{\mathbb{D}}|f(z)|^p(1-|z|^2)^{2+\alpha} d\mu(z)\leq \Lambda\|f\|^p_{p,\alpha} \] for all \(f\in A^p(\alpha)\). (The notion of a sampling measure is a continuous analogue of sets of sampling discussed by K. Seip and A. Schuster.) This paper provides a characterization of sampling measures for \(A^p(\alpha)\). The characterization is in terms of weak\(^*\) limits of Moebius transforms of the measure and reads as follows: Let \(M\) be the set of Moebius transforms \(\mathbb{D}\), and, for a fixed measure \(\mu\), \(\mu(M)= \{\mu_\phi: \phi\in M\}\) where \(\mu_\phi(E)= \mu(\phi^{-1}(E))\), \(E\) a measurable subset of \(\mathbb{D}\). Finally, let \(W_\mu\) be the set of all weak limits of measures in \(\mu(M)\), where weak limit is defined in the standard way. Then: Theorem: Let \(\mu\in C\). The following are equivalent (a) \(\mu\) is a sampling measure for \(A^p(\alpha)\). (b) There exists a \(q<p\) such that the support of every measure in \(W_\mu\) is a set of uniqueness for \(A^q(\alpha)\) [that is, a set which fails to be the zero set for any \(f\in A^q(\alpha)\)]. (c) There exists a \(\beta< \alpha\) such that the support of every measure in \(W_\mu\) is a set of uniqueness for \(A^p(\beta)\). The paper has much expository and mathematical work relating this result to earlier results of the author and the contemporary work of others, especially K. Seip and A. Schuster. There is a long final section on ``further considerations'' which place this theorem in the context of similar work by others, and raises the (at present) unsolved issue of characterizing these measures by a density criteria, as has been done for sampling sets by Seip and Schuster. A theorem along these lines is given.
scientific article; zbMATH DE number 1463208