Boundary Values of Harmonic Functions in Spaces of Triebel-Lizorkin Type

Chin Cheng Lin, Ying Chieh Lin

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Abstract

Triebel (J Approx Theory 35:275-297, 1982; 52:162-203, 1988) investigated the boundary values of the harmonic functions in spaces of the Triebel-Lizorkin type Fα, qp on Rn+1+ by finding an characterization of the homogeneous Triebel-Lizorkin space Ḟα,qp via its harmonic extension, where 0 < p < ∞, 0 < q ≤ ∞, and α < min{-n/p, -n/q}. In this article, we extend Triebel's result to α < 0 and 0 < p, q ≤ ∞ by using a discrete version of reproducing formula and discretizing the norms in both Ḟα,qp. Furthermore, for α < 0 and 1 < p,q ≤ ∞, the mapping from harmonic functions in Fα,qp to their boundary values forms a topological isomorphism between Fα,qp and Ḟα,qp.

Original languageEnglish
Pages (from-to)23-48
Number of pages26
JournalIntegral Equations and Operator Theory
Volume79
Issue number1
DOIs
StatePublished - May 2014

Keywords

  • Boundary values
  • harmonic functions
  • Triebel-Lizorkin spaces

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