258 research outputs found
Holomorphic extension from the sphere to the ball
Real analytic functions on the boundary of the sphere which have separate
holomorphic extension along the complex lines through a boundary point have
holomorphic extension to the ball. This was proved in a previous preprint by an
argument of CR geometry. We give here an elementary proof based on the
expansion in holomorphic and antiholomorphic powers
Propagation of boundary CR foliations and Morera type theorems for manifolds with attached analytic discs
We prove that generic homologically nontrivial -parameter family of
analytic discs attached by their boundaries to a CR manifold in
tests CR functions: if a smooth function on
extends analytically inside each analytic disc then it satisfies the tangential
CR equations.
In particular, we answer, in real analytic category, two open questions: on
characterization of analytic functions in planar domains (the strip-problem),
and on characterization of boundary values of holomorphic functions in domains
in (a conjecture of Globevnik and Stout). We also characterize
complex curves in as real 2-manifolds admitiing homologically
nontrivial 1-parameter families of attached analytic discs.
The proofs are based on reduction to a problem of propagation of degeneracy
of CR foliations of torus-like manifolds.Comment: The version accepted in Advances in Mathematic
Malmheden's theorem revisited
In 1934 H. Malmheden discovered an elegant geometric algorithm for solving
the Dirichlet problem in a ball. Although his result was rediscovered
independently by Duffin 23 years later, it still does not seem to be widely
known. In this paper we return to Malmheden's theorem, give an alternative
proof of the result that allows generalization to polyharmonic functions and,
also, discuss applications of his theorem to geometric properties of harmonic
measures in balls in Euclidean spaces
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