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 \({\mathcal F^{\alpha,q}_{p}}\) on \({\mathbb{R}^{n+1}_+}\) by finding an characterization of the homogeneous Triebel–Lizorkin space \({{\bf \dot{F}}^{\alpha,q}_p}\) via its harmonic extension, where \({0 < p < \infty, 0 < q \leq \infty}\), and \({\alpha < {\rm min}\{-n/p, -n/q\}}\). In this article, we extend Triebel’s result to α < 0 and \({0 < p, q \leq \infty}\) by using a discrete version of reproducing formula and discretizing the norms in both \({\mathcal{F}^{\alpha,q}_{p}}\) and \({{\bf{\dot{F}}}^{\alpha,q}_p}\). Furthermore, for α < 0 and \({1 < p,q \leq \infty}\), the mapping from harmonic functions in \({\mathcal{F}^{\alpha,q}_{p}}\) to their boundary values forms a topological isomorphism between \({\mathcal{F}^{\alpha,q}_{p}}\) and \({{\bf \dot{F}}^{\alpha,q}_p}\).
Similar content being viewed by others
References
Bownik M.: Duality and interpolation of anisotropic Triebel–Lizorkin spaces. Math. Z. 259, 131–169 (2008)
Bui H.-Q.: Harmonic functions, Riesz potentials, and the Lipschitz spaces of Herz. Hiroshima Math. J. 9, 245–295 (1979)
Bui H.-Q.: Characterizations of weighted Besov and Triebel–Lizorkin spaces via temperatures. J. Funct. Anal. 55, 39–62 (1984)
Bui H.-Q.: Representation theorems and atomic decomposition of Besov spaces. Math. Nachr. 132, 301–311 (1987)
Coifman R., Rochberg R.: Representation theorems for holomorphic and harmonic functions in L p. Asterisque 77, 12–66 (1980)
Fefferman C., Stein E.M.: H p spaces of several variables. Acta Math. 129, 137–193 (1972)
Frazier M., Jawerth B.: Decomposition of Besov Spaces. Indiana Univ. Math. J. 34, 777–799 (1985)
Frazier M., Jawerth B.: A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93, 34–170 (1990)
Han Y., Lee M.-Y., Lin C.-C.: Hardy spaces and the Tb theorem. J. Geom. Anal. 14, 291–318 (2004)
Jawerth B.: Some observation on Besov and Lizorkin–Triebel spaces. Math. Scand. 40, 94–104 (1977)
Ricci F., Taibleson M.: Boundary values of harmonic functions in mixed norm spaces and their atomic structure. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10, 1–54 (1983)
Triebel H.: Characterizations of Besov–Hardy–Sobolev spaces via harmonic functions, temperatures and related means. J. Approx. Theory 35, 275–297 (1982)
Triebel H.: Theory of Function Spaces. Monographs in Mathematics, vol. 78. Birkhäuser, Basel (1983)
Triebel H.: Characterizations of Besov–Hardy–Sobolev spaces: a unified approach. J. Approx. Theory 52, 162–203 (1988)
Wang K.: The full T1 theorem for certain Triebel–Lizorkin space. Math. Nachr. 197, 103–133 (1999)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lin, CC., Lin, YC. Boundary Values of Harmonic Functions in Spaces of Triebel–Lizorkin Type. Integr. Equ. Oper. Theory 79, 23–48 (2014). https://doi.org/10.1007/s00020-014-2137-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00020-014-2137-x