Local a priori estimates in L-p for first order linear operators with nonsmooth coefficients

We prove local a priori estimates in L-P, l<p<infinity, for first-order 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 the coefficients is assumed. This extends to L-P spaces estimates that were previously known for p=2. Examples show that the regularity required from the coefficients is essentially minimal.