Displaying similar documents to “Boundary functions in L 2 H ( 𝔹 n ) ”

Boundary functions on a bounded balanced domain

Piotr Kot (2009)

Czechoslovak Mathematical Journal

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We solve the following Dirichlet problem on the bounded balanced domain Ω with some additional properties: For p > 0 and a positive lower semi-continuous function u on ∂ Ω with u ( z ) = u ( λ z ) for | λ | = 1 , z ∈ ∂ Ω we construct a holomorphic function f ∈ 𝕆 ( Ω ) such that u ( z ) = ∫ 𝔻 z | f | p d 𝔏 𝔻 z 2 for z ∈ ∂ Ω , where 𝔻 = { λ ∈ ℂ | λ | < 1 } .

The Lindelöf principle in ℂn

Peter Dovbush (2013)

Open Mathematics

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Let D be a bounded domain in ℂn. A holomorphic function f: D → ℂ is called normal function if f satisfies a Lipschitz condition with respect to the Kobayashi metric on D and the spherical metric on the Riemann sphere ̅ℂ. We formulate and prove a few Lindelöf principles in the function theory of several complex variables.

Description of simple exceptional sets in the unit ball

Piotr Kot (2004)

Czechoslovak Mathematical Journal

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For z ∈ ∂ B n , the boundary of the unit ball in ℂ n , let Λ ( z ) = { λ | λ | ≤ 1 } . If f ∈ 𝕆 ( B n ) then we call E ( f ) = { z ∈ ∂ B n ∫ Λ ( z ) | f ( z ) | 2 d Λ ( z ) = ∞ } the exceptional set for f . In this note we give a tool for describing such sets. Moreover we prove that if E is a G δ and F σ subset of the projective ( n - 1 ) -dimensional space ℙ n - 1 = ℙ ( ℂ n ) then there exists a holomorphic function f in the unit ball B n so that E ( f ) = E .