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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
##### Keywords:
Carleson measures; Bergman spaces; Dirichlet spaces; multipliers
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