Closed Range and Essential Norms of Composition Operators on Weighted Dirichlet Spaces

Abstract

We study closed range and essential norms of bounded composition operators $C_φ$ on weighted Dirichlet spaces $\mathcal{D}_α$. For $-1<α<0$, we first consider maps whose images are obtained by removing a compact subset from a simply connected subdomain of $\mathbb{D}$. In this setting, closed range is equivalent to the Reverse Carleson property and the Reverse Carleson Disk Condition for the counting measure $μ_{φ,α}$, uniform weighted area density of the image, $\mathbb{T}\subset\overline{φ(\mathbb{D})}$, and the inclusion of an outer annulus in the image. We also obtain a closed range characterization under the Uniform Tail Condition. For all $α>-1$, we establish two-sided essential norm estimates in terms of local averages of the generalized Nevanlinna counting function and an exact formula using boundary tail integrals. If the normalized counting density has vanishing oscillation at the boundary, we obtain exact formulas in terms of its boundary limsup, Berezin transform, and local averages.