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

We study the maximal function M f ( 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 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   α p,∞(R N ) into L p (R N ). The admissible regions for the spaces B   α p,q(R N ) with p < q are more complicated.