LOCAL A-PRIORI ESTIMATES IN L-P FOR FIRST-ORDER LINEAR-OPERATORS WITHNONSMOOTH COEFFICIENTS

We prove local a priori estimates in L-P, l<p<infinity, for first-orde r linear operators that satisfy the Nirenberg-Treves condition (P) and whose coefficients have Lipschitz continuous derivatives of order one . When the number of variables is two, only Lipschitz continuity of th e coefficients is assumed. This extends to L-P spaces estimates that w ere previously known for p=2. Examples show that the regularity requir ed from the coefficients is essentially minimal.