Boundary Behaviour of Functions in Weighted Dirichlet Spaces

We consider the class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{D}_W$$\end{document} of holomorphic functions f(z) = Σ anzn in the unit disc for which Σ W(n)|an|2 < ∞, where the weight function W satisfies standard regularity conditions. We show that if Σ 1/(nW(n)) < ∞ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f \in \mathcal{D}_W$$\end{document}, then the radial variation Lf(θ) = ∫01|f′(reiθ)| 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{D}_W$$\end{document} with Lf(θ) = ∞ for every θ. We also show that it is a consequence of known results that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f \in \mathcal{D}_W$$\end{document} and Σ 1/W(n) = ∞, then f has finite radial, and non-tangential, limits outside certain exceptional sets.