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Carleson measures for spaces of Dirichlet type. (English) Zbl 1103.30035
Let \(\mathbb{D}\) be the unit disc \(\{z\in\mathbb{C}:| z| <1\}\) and let \({\mathcal Hol}\mathbb{(D)}\) be the space of all analytic functions in \(\mathbb{D}\). For \(0<p<\infty\) and \(\alpha>-1\), let \[ \mathcal{D}_\alpha^{p}=\{f\in{\mathcal H}ol\,\mathbb{D}:f'\in A_\alpha^{p}\}, \] where \(A_\alpha^{p}\) is the weighted Bergman space \[ \biggl\{g\in{\mathcal Hol}(\mathbb{D}):\int_{\mathbb{D}}(1-| z| )^{\alpha}| g(z)| ^{p}\,dA(z)<\infty\biggr\}. \] A positive Borel measure \(\mu\) in \(\mathbb{D}\) is a classical Carleson measure if there exists a positive constant \(C\) such that \(\mu\left(S(I)\right)\leq C| I| \) for every interval \(I\subset\mathbb{T}=\partial\mathbb{D}\), where \(| I| \) is the length of \(I\) and \(S(I)\) is the Carleson box. Further, a positive Borel measure \(\mu\) in \(\mathbb{D}\) is said to be a Carleson measure for \(\mathcal{D}_\alpha^{p}\) if \(\mathcal{D}_\alpha^{p}\subset L^{p}(d\mu)\). A characterization of the Carleson measures for \(\mathcal{D}_\alpha^{p}\) for certain values of \(p\) and \(\alpha\) was proved by Z. Wu [J. Funct. Anal. 169, No.1, 148–163 (1999; Zbl 0962.30032)]. In particular, he proved that, for \(0<p\leq2\), the Carleson measures for the space \(\mathcal{D}_{p-1}^{p}\) are precisely the classical Carleson measures. Wu conjectured that this result remains true for \(2<p<\infty\). The authors in the paper under review prove that this conjecture is false. In fact, they prove that, for \(2<p<\infty\), there exists \(g\in{\mathcal Hol}(\mathbb{D})\) such that the measure \(\mu_{g,p}\) on \(\mathbb{D}\) defined by \[ d\mu_{g,p}(z)=(1-| z| ^{2})^{p-1}| g'(z)| ^{p}dA(z) \] is not a Carleson measure for \(\mathcal{D}_{p-1}^{p}\) but is a classical Carleson measure. Also some sufficient conditions for multipliers of the spaces \(\mathcal{D}_{p-1}^{p}\) are given.

MSC:
30H05 Spaces of bounded analytic functions of one complex variable
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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