Computational Methods and Function Theory 7 (2007), No. 2, 361--370
Copyright Heldermann Verlag 2007
Finbarr Holland
f.holland@ucc.ie , University College Cork, Department of Mathematics, Cork, Ireland.
J. Brian Twomey
b.twomey@ucc.ie , University College Cork, Department of Mathematics, Cork, Ireland.
We consider the class DW of holomorphic functions f(z) = Σ an zn 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 ∈ DW, then the radial variation Lf(θ) = ∫01 |f'(r ei θ)| 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 DW with Lf(θ) = ∞ for every θ. We also show that it is a consequence of known results that if f ∈ DW and Σ 1/W(n) = ∞, then f has finite radial, and non-tangential, limits outside certain exceptional sets.
Keywords: Weighted Dirichlet spaces, radial variation, radial limits, exceptional sets, zero capacity.
MSC 2000: 30H05, 31A20.
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