Tangential Boundary Behaviour of Bounded Harmonic Functions in the Unit Disc

Fausto Di Biase, Alexander M. Stokolos, Olof Svensson, Tomasz Weiss

Research output: Contribution to book or proceedingChapter

Abstract

Bounded harmonic functions in the unit disc D converge nontangentially almost everywhere (Fatou, 1906) and fail to converge along the rotates of any given tangential curve (Littlewood, 1927). We study their boundary behaviour along tangential curves whose shape may change from point to point (a problem posed by W. Rudin). Let τ be the assignment of a curve τ θ in D ending at θ and tangential to the boundary bD of D, for each θ∈bD. The authors announce a proof that convergence along τ fails if τ θ depends on θ in a measurable way, and to show that there is a family τ of tangential curves such that each bounded harmonic function in D converges along τ θ for a set of points θ whose outer measure is equal to 2π. Proofs and further extensions of their theorems 1 and 2 will be given elsewhere.
Original languageAmerican English
Title of host publicationSeminars of Geometry, University of Bologna, Italy, 1996–1997
StatePublished - 1998

Disciplines

  • Mathematics

Keywords

  • Bounded harmonic functions
  • Tangential boundary behavior
  • Unit disc

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