L1 Sobolev estimates for (pseudo)-differential operators and applications

In this work we show that if A(x,D) is a linear differential operator of order with smooth complex coefficients in RN from a complex vector space E to a complex vector space F, the Sobolev a priori estimate holds locally at any point x0 if and only if A(x,D) is elliptic and the constant coefficient homogeneous operator A(x0,D) is canceling in the sense of Van Schaftingen for every x0 which means that >Here A(x,D) is the homogeneous part of order of A(x,D) and a(x,) is the principal symbol of A(x,D). This result implies and unifies the proofs of several estimates for complexes and pseudo-complexes of operators of order one or higher proved recently by other methods as well as it extends in the local setup the characterization of Van Schaftingen to operators with variable coefficients.