On local continuous solvability of equations associated to elliptic and canceling linear differential operators

Abstract

Consider A(x,D):C∞(Ω,E)→C∞(Ω,F) an elliptic and canceling linear differential operator of order ν with smooth complex coefficients in Ω⊂RN from a finite dimension complex vector space E to a finite dimension complex vector space F and A⁎(x,D) its adjoint. In this work we characterize the (local) continuous solvability of the partial differential equation A⁎(x,D)v=f (in the distribution sense) for a given distribution f; more precisely we show that any x0∈Ω is contained in a neighborhood U⊂Ω in which its continuous solvability is characterized by the following condition on f: for every ε>0 and any compact set K⊂⊂U, there exists θ=θ(K,ε)>0 such that the following holds for all smooth function φ supported in K:|f(φ)|⩽θ‖φ‖Wν−1,1+ε‖A(x,D)φ‖L1, where Wν−1,1 stands for the homogenous Sobolev space of all L1 functions whose derivatives of order ν−1 belongs to L1(U).This characterization implies and extends results obtained before for operators associated to elliptic complexes of vector fields (see [1]); we also provide local analogues, for a large range of differential operators, to global results obtained for the classical divergence operator by Bourgain and Brezis in [2] and by De Pauw and Pfeffer in [3].