Carleson measures and multipliers for Dirichlet spaces
The Dirichlet space \(\mathcal D^p_\alpha\) consists of those analytic functions on the unit disk whose derivative belongs to the Bergman space \(\mathcal A^p_\alpha\) of functions \(f\) such that \(\int_{\mathbb D} |f(z)|^p(1-|z|^2)^\alpha dA\) is finite, where \(dA\) is a normalized area measure on the disk. When \(p\geq \alpha+1\) the Dirichlet space is not itself a Bergman space and is best viewed in relation to analytic extensions of elements of the Besov space \(\mathcal B_\sigma^p\) consisting of functions that are in \(L^p\) of the unit circle and such that \(\int_0^{2\pi} \int_0^{2\pi} \frac{|f(\theta+t)-f(\theta)|^p}{|t|^{1+p\sigma} } d\theta dt\) is finite. A Carleson measure for \(\mathcal D^p_\alpha\) is a continuous functional on \(\mathcal D^p_\alpha\) whose action on \(g\in \mathcal D^p_\alpha\) is defined by integration of \(|g(z)|^p\) over the disk against an appropriate nonnegative measure \(\mu\). The author characterizes such Carleson measures. In the most interesting case in which \(\min(1,\alpha)<p\leq \alpha +2\) the Carleson measures are characterized by the condition that, for any open subset \(\mathcal O\) of the unit circle one has \(\mu(T(\mathcal O)) \leq C\text{ cap}(\mathcal O;\mathcal B^p_{1-(\alpha+1)/p})\). Here \(T(\mathcal O)\) is the tent over \(\mathcal O\) and \(\text{ cap}(\mathcal O;X)= \inf\{ \|f\|_X:f\chi_{\mathcal O}\geq \chi_{\mathcal O} \}\). For other values of \(\alpha,p\) the Carleson measures are characterized more readily in terms of \(\mu(T(I))\) being bounded by a power of \(|I|\) where \(I\) is a subinterval of the circle. The characterizations of Carleson measures are used to characterize the pointwise multipliers of \(\mathcal D^p_\alpha\) as those \(f\in H^{\infty}\) of the disk such that \(|f'|^p(1-|z|^2)^\alpha dA\) is a Carleson measure.
scientific article; zbMATH DE number 1398376
scientific article