Carleson Measures for Spaces of Dirichlet Type

Abstract.

If 0  <  p  <  ∞ and α  >   − 1, the space $\mathcal{D}_\alpha ^p$ consists of those functions f which are analytic in the unit disc $\mathbb{D}$ and have the property that f ′ belongs to the weighted Bergman space A ^p_α . In 1999, Z. Wu obtained a characterization of the Carleson measures for the spaces $\mathbb{D}_\alpha ^p$ for certain values of p and α. In particular, he proved that, for 0  <  p ≤ 2, the Carleson measures for the space $\mathbb{D}_{p - 1}^p$ 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 $\mathbb{D}$ such that the measure μ_g,p on $\mathbb{D}$ defined by dμ_g,p (z)  =  (1  −  |z|²)^{p - 1}| g ′ (z)|^p dx dy is not a Carleson measure for $\mathcal{D}_{p - 1}^p$ but is a classical Carleson measure. We obtain also some sufficient conditions for multipliers of the spaces $\mathcal{D}_{p - 1}^p .$