Carleson Measures for the Bloch Space

Abstract.

In this paper we study the positive Borel measures μ on the unit disc ${\mathbb{D}}$ in ${\mathbb{C}}$ for which the Bloch space $\mathcal{B}$ is continuously included in $L^p(d\mu)$, 0 < p < ∞. We call such measures p-Bloch-Carleson measures. We give two conditions on a measure μ in terms of certain logarithmic integrals one of which is a necessary condition and the other a sufficient condition for μ being a p-Bloch-Carleson measure. We also give a complete characterization of the p-Bloch-Carleson measures within certain special classes of measures. It is also shown that, for p > 1, the p-Bloch-Carleson measures are exactly those for which the Toeplitz operator $T_\mu$, defined by $T_\mu(f)(z) = \int_\mathbb{D} {\frac {f(w)} {(1-\bar{w}z)^2}} d\mu(w) (f \epsilon L^1(d\mu), z \epsilon {\mathbb{D}})$, maps continuously $L^{p\prime}\,(d\mu)$ into the Bergman space A ¹, $\frac {1} {p}\,+\,\frac {1}{p\prime}\,=\,1$. Furthermore, we prove that if p > 1, α >-1 and ω is a weight which satisfies the Bekollé-Bonami $\mathcal{B}_{p,\alpha}$-condition, then the measure $\mu_{\alpha,p}$ defined by $d\mu_{\alpha,p}(z) = {(1-|z|^2)}^{\alpha}\omega(z)dA(z)$ is a p-Bloch-Carleson-measure.

We also consider the Banach space $H^{\infty}_{\rm log}$ of those functions f which are analytic in ${\mathbb{D}}$ and satisfy $|f(z)| = O\left({\rm log} \frac {1} {1-|z|}\right)$, as $|z| \rightarrow 1$. The Bloch space is contained in $H^{\infty}_{\rm log}$. We describe the p-Carleson measures for $H^{\infty}_{\rm log}$ and study weighted composition operators and a class of integration operators acting in this space. We determine which of these operators map $H^{\infty}_{\rm log}$ continuously to the weighted Bergman space $A^{p}_{\alpha} (p > 0, \alpha > -1)$ and show that they are automatically compact.