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 omega is a region in the [GRAPHICS] upper half space R+N+1 and f(x,t) is the ha rmonic extension to R+N+1 of a distribution in the Besov space B(p,q)a lpha(R(N)) or in the Triebel-Lizorkin space F(p,q)alpha(R(N)). In part icular, we prove that when OMEGA = {\y\N/(N-alphap) < t < 1} the opera tor M is bounded from F(p,infinity)alpha(R(N)) into L(p)(R(N)). The ad missible regions for the spaces B(p,q)alpha(R(N)) with p < q are more complicated.