Carleson Measures for Spaces of Dirichlet Type

If 0  <  p  <  ∞ and α  >   − 1, the space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{D}_\alpha ^p $$\end{document} consists of those functions f which are analytic in the unit disc \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{D}$$\end{document} and have the property that f ′ belongs to the weighted Bergman space Aαp. In 1999, Z. Wu obtained a characterization of the Carleson measures for the spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{D}_\alpha ^p$$\end{document} for certain values of p and α. In particular, he proved that, for 0  <  p ≤ 2, the Carleson measures for the space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{D}_{p - 1}^p$$\end{document} are precisely the classical Carleson measures. Wu also conjectured that this result remains true for 2  <  p  <  ∞. In this paper we prove that this conjecture is false. Indeed, we prove that if 2  <  p  <  ∞, then there exists g analytic in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{D}$$\end{document} such that the measure μg,p on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{D}$$\end{document} defined by dμg,p (z)  =  (1  −  |z|2)p - 1| g ′ (z)|p dx dy is not a Carleson measure for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{D}_{p - 1}^p$$\end{document} but is a classical Carleson measure. We obtain also some sufficient conditions for multipliers of the spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{D}_{p - 1}^p .$$\end{document}