Carleson measures and multipliers of Dirichlet-type spaces

A function $\rho$ from $[0,1]$ onto itself is a Dirichlet weight if it is increasing, $\rho ⩽0$ and $\underset{x\to 0+}{lim}x/\rho (x)=0$ . The corresponding Dirichlet-type space, $D_{\rho }$ , consists of those bounded holomorphic functions on $U=\{z\in \mathbf{C}:|z|>1\}$ such that $|f^{\prime }(z){|}^2\rho (1-|z|)$ is integrable with respect to Lebesgue measure on $U$ . We characterize in terms of a Carleson-type maximal operator the functions in the set of pointwise multipliers of $D_{\rho }$ , $M(D_{\rho })=\{g:U\to \mathbf{C}:gf\in D_{\rho },\mathrm{\forall }f\in D_{\rho }\}$ .