Boundary Behaviour of Functions in Weighted Dirichlet Spaces

Abstract

We consider the class \(\mathcal{D}_W\) of holomorphic functions f(z) = Σ a n z n in the unit disc for which Σ W(n)|a n |2 < ∞, where the weight function W satisfies standard regularity conditions. We show that if Σ 1/(nW(n)) < ∞ and \(f \in \mathcal{D}_W\), then the radial variation L f (θ) = ∫ 0 1 |f′(re iθ )| dr is finite outside an exceptional set of capacity zero, where the kernel associated with the capacity depends on W. It is known that if Σ 1/(nW(n)) = ∞, then there exist functions in \(\mathcal{D}_W\) with L f(θ) = ∞ for every θ. We also show that it is a consequence of known results that if \(f \in \mathcal{D}_W\) and Σ 1/W(n) = ∞, then f has finite radial, and non-tangential, limits outside certain exceptional sets.