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 Fpα,q\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal F^{\alpha,q}_{p}}$$\end{document} on R+n+1\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb{R}^{n+1}_+}$$\end{document} by finding an characterization of the homogeneous Triebel–Lizorkin space F˙pα,q\documentclass[12pt]{minimal}
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\begin{document}$${{\bf \dot{F}}^{\alpha,q}_p}$$\end{document} via its harmonic extension, where 0<p<∞,0<q≤∞\documentclass[12pt]{minimal}
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\begin{document}$${0 < p < \infty, 0 < q \leq \infty}$$\end{document}, and α<min{-n/p,-n/q}\documentclass[12pt]{minimal}
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\begin{document}$${\alpha < {\rm min}\{-n/p, -n/q\}}$$\end{document}. In this article, we extend Triebel’s result to α < 0 and 0<p,q≤∞\documentclass[12pt]{minimal}
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\begin{document}$${0 < p, q \leq \infty}$$\end{document} by using a discrete version of reproducing formula and discretizing the norms in both Fpα,q\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{F}^{\alpha,q}_{p}}$$\end{document} and F˙pα,q\documentclass[12pt]{minimal}
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\begin{document}$${{\bf{\dot{F}}}^{\alpha,q}_p}$$\end{document}. Furthermore, for α < 0 and 1<p,q≤∞\documentclass[12pt]{minimal}
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\begin{document}$${1 < p,q \leq \infty}$$\end{document}, the mapping from harmonic functions in Fpα,q\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{F}^{\alpha,q}_{p}}$$\end{document} to their boundary values forms a topological isomorphism between Fpα,q\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{F}^{\alpha,q}_{p}}$$\end{document} and F˙pα,q\documentclass[12pt]{minimal}
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\begin{document}$${{\bf \dot{F}}^{\alpha,q}_p}$$\end{document}.
