Carleson measures and multipliers of Dirichlet-type spaces

Abstract:

A function $\rho$ from $[0, 1]$ onto itself is a Dirichlet weight if it is increasing, $\rho '' \leqslant 0$ and ${\lim _{x \to 0 + }}x/\rho (x) = 0$. The corresponding Dirichlet-type space, ${D_\rho }$, consists of those bounded holomorphic functions on $U = \{ z \in {\mathbf {C}}: |z| < 1\}$ such that $|f’(z){|^2}\rho (1 - |z|)$ 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 ${D_\rho }$, $M({D_\rho }) = \{ g: U \to {\mathbf {C}}:gf \in {D_\rho },\forall f \in {D_\rho }\}$.