Tangential Convergence of Temperatures and Harmonic Functions in Besov and in Triebel-Lizorkin Spaces

We study the maximal function Mf(x) = sup |f(x + y, t)| when Ω is a region in the (y,t) Ω upper half space R and f(x, t) is the harmonic extension to R₊^N+1 of a distribution in the Besov space B^α_p,q(R^N) or in the Triebel-Lizorkin space F^α_p,q(R^N). In particular, we prove that when Ω= {|_y|^{N/ (N-αp)} < t < 1} the operator M is bounded from F (R^N) into L^p (R^N). The admissible regions for the spaces B (R^N) with p < q are more complicated.