Boundary Value Problems for Elliptic Equations

where án, denotes differentiation in the direction of tlie normal to 8B . As is well known, there are explicit formulas for the solutions of the aboye problems, and one can then give a very careful analysis of the solutions when, say f E LP(áB, do), 1 < p < oo . In both cases, the boundary values are taken in the sense of non-tangential convergence, Le ., if Q E aB, and F(Q) _ F. (Q) _ {X E B1 IX Q1 < (1 + a)dist(X,áB)}, a > 0, in the case of (D) we have lim x_Q u(X) = f(Q) for a.e . Q(dv), and in the case of (N) XEr(4) we have limX_Q Vu(X)NQ = f(Q) for a.e . Q(da) . Moreover, the HardyXEr(4) Littlewood maximal theorem, and in the case of (N), this theorem and the Calderón-Zygmund theory of singular integrals give, denoting by u*(Q) _ supxEr(Q) lu(X)j, the estimates