Author's Department

Mathematics

Document Type

Article

Publication Date

2012

Abstract

If φ is an analytic function bounded by 1 on the bidisk D 2 and τ∈∂(D 2 ) is a point at which φ has an angular gradient ∇φ(τ) then ∇φ(λ)→∇φ(τ) as λ → τ nontangentially in D 2 . This is an analog for the bidisk of a classical theorem of Carathéodory for the disk. For φ as above, if τ∈∂(D 2 ) is such that the lim inf of (1−|φ(λ)|)/(1−∥λ∥) as λ → τ is finite then the directional derivative Dδφ(τ) exists for all appropriate directions δ∈C 2 . Moreover, one can associate with φ and τ an analytic function h in the Pick class such that the value of the directional derivative can be expressed in terms of h.

Comments

Author version. Original publication in Mathematische Annalen Volume 352, Issue 3 , pp 581-624 is available at link.springer.com. DOI: 10.1007/s00208-011-0650-7.

Embargo Period

1-3-2013

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