Square roots of elliptic second order divergence operators on strongly Lipschitz domains: L-p theory

We study L-P estimates for square roots of second order elliptic non necess arily selfadjoint operators in divergence form L = - div (A del) on Lipschi tz domains subject to Dirichlet or to Neumann boundary conditions, pursuing our work [4] where we considered operators on R". We obtain among other th ings parallel toL(1/2)f parallel top less than or equal to c parallel to de lf parallel top for all 1 < p < infinity if L is real symmetric and the dom ain bounded, which is new for 1 < p < 2. We also obtain similar results for perturbations of constant coefficients operators. Our methods rely on a si ngular integral representation, Calderon-Zygmund theory and quadratic estim ates. A feature of this study is the use of a commutator between the resolv ent of the Laplacian (Dirichlet and Neumann) and partial derivatives which carries the geometry of the boundary.