ABSTRACT
R3 . We obtain the optimal estimates for the sharp ranges of LP spaces. The corresponding
results for p = 2 in Rn , n ~ 3, were obtained in (Dahlberg, et al., 1988). The extension of the results in the present paper to the case of Rn, n ~ 4 remains an open problem. The corresponding theorems for the Dirichlet and Neumann problem for Laplace's equation in
a Lipschitz domain are by now well understood in all dimensions, thanks to the results in
(Dahlberg, 1979), (Jerison-Kenig, 1981), (Verchota, 1984), and (Dahlberg-Kenig, 1987).
