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 N+1+ and f(x, t) is the harmonic extension to R+N+1 of a distribution in the Besov space Bαp,q(RN) or in the Triebel‐Lizorkin space Fαp,q(RN). In particular, we prove that when Ω= {|y|N/ (N‐αp) < t < 1} the operator M is bounded from F αp,∞ (RN) into Lp (RN). The admissible regions for the spaces B αp,q (RN) with p < q are more complicated