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Boundary cross theorem in dimension 1

Peter Pflug, Viêt-Anh Nguyên (2007)

Annales Polonici Mathematici

Let X, Y be two complex manifolds of dimension 1 which are countable at infinity, let D ⊂ X, G ⊂ Y be two open sets, let A (resp. B) be a subset of ∂D (resp. ∂G), and let W be the 2-fold cross ((D∪A)×B) ∪ (A×(B∪G)). Suppose in addition that D (resp. G) is Jordan-curve-like on A (resp. B) and that A and B are of positive length. We determine the "envelope of holomorphy" Ŵ of W in the sense that any function locally bounded on W, measurable on A × B, and separately holomorphic on (A × G) ∪ (D × B)...

Boundary functions in L 2 H ( 𝔹 n )

Piotr Kot (2007)

Czechoslovak Mathematical Journal

We solve the Dirichlet problem for line integrals of holomorphic functions in the unit ball: For a function u which is lower semi-continuous on ∂ 𝔹 n we give necessary and sufficient conditions in order that there exists a holomorphic function f ∈ 𝕆 ( 𝔹 n ) such that u ( z ) = ∫ | λ | < 1 f ( λ z ) 2 d 𝔏 2 ( λ ) .

Boundary regularity of admissible operators.

Christoph H. Lampert (2005)

Publicacions Matemàtiques

In strictly pseudoconvex domains with smooth boundary, we prove a commutator relationship between admissible integral operators, as introduced by Lieb and Range, and smooth vector fields which are tangential at boundary points. This makes it possible to gain estimates for admissible operators in function spaces which involve tangential derivatives. Examples are given under with circumstances these can be transformed into genuine Sobolev- and Ck-estimates.

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