<i>L</i>^<i>p</i>-integrability of the second order derivatives of Green potentials in convex domains

Abstract

We give estimates in LP , 1 < p < 2, of the second order derivatives of the Green potential of / e L p , for convex domains.This is done by interpolating between estimates in L 1 and L 2 of functions in atomic H ι and L 2 , 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.O Introduction.The purpose of this paper is to prove that for a convex domain Ω in R n , the second order derivatives of the Green potential of /, V 2 Gf, are in LP(Ω) if / e LP(Ω) with 1 < p < 2. To avoid technicalities we assume throughout the paper that n > 3. The main results of the paper are