Fatou Theorems on Domains in C

for a.e. P G dD. It is known [IP] that this nontangential method of approach is best possible. If O Ç C n is a smoothly bounded domain then there is a similar theory of H spaces (also classical). (Let 6Q(Z) = dist(z,dQ).) In this theory one replaces (i) the circles {re: 0 < 9 < 2TT} by dQe = {z G H: 6Q(Z) = e}, e small; (ii) linear measure by (2n — l)-dimensional area measure; (iii) Stolz regions Ta by cones in space of fixed aperture. It is a remarkable discovery of Korânyi (for the ball and for certain symmetric domains [AKI, AK2]) and of Stein (for smoothly bounded domains [ESI]) that in C n , n > 1, the conical approach regions are not optimal for studying the boundary behavior of H functions. Indeed, they may be replaced by admissible approach regions which are conical in "complex normal directions" and parabolic in "complex tangential directions" (see [ESI, SK]).