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

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 F^{α, q}_p on Rⁿ⁺¹₊ by finding an characterization of the homogeneous Triebel-Lizorkin space Ḟ^α,q_p 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 Ḟ^α,q_p. Furthermore, for α < 0 and 1 < p,q ≤ ∞, the mapping from harmonic functions in F^α,q_p to their boundary values forms a topological isomorphism between F^α,q_p and Ḟ^α,q_p.