Kato's Inequalities for Degenerate Quasilinear Elliptic Operators

Let N >= 1 and p > 1. Let Omega be a domain of RN. In this article we shall establish Kato's inequalities for quasilinear degenerate elliptic operators of the form A(p)u = divA(x, Vu) for u epsilon K-p(Q), where K-p(Q) is an admissible class and A(x,) : Q x R-N > R-N is a mapping satisfying some structural conditions. If p = 2 for example, then we have K-2(Q) = {u epsilon L-loc(1)(Q) : 0,u, epsilon LL,(Q) for j, k = 1,2, ..., N}. Then we shall prove that Ap vertical bar u vertical bar > (sgn u) Apu and A(p)u + > (sgn(+) u)(P-1) A(p)u in D'(Q) with u epsilon Kp(Q). These inequalities are called Kato's inequalities provided that p = 2. The class of operators A(p) contains the so-called p harmonic operators L-p = diva (vertical bar del u vertical bar P-2V del u) for A(x, xi) = vertical bar xi vertical bar(p-2) xi.