L(P)-INTEGRABILITY OF THE 2ND-ORDER DERIVATIVES OF GREEN POTENTIALS IN CONVEX DOMAINS

We give estimates in L(p), 1 < p less-than-or-equal-to 2, of the second order derivatives of the Green potential of f is-an-element-of L(p), for convex domains. This is done by interpolating between estimates in L1 and L2 of functions in atomic H-1 and L2, respectively. The crucial step is obtaining the atomic estimate which is done by adapting to the present situation, a technique introduced by Dahlberg and Kenig.