Some results about the structure of primitivity domains for linear partial differential operators with constant coefficients.

Let G(D) be a linear partial differential operator on Rn, with constant coefficients. Moreover let Ω⊂Rn be open and F∈L1loc(Ω,CN). Then any set of the form Af,F:={x∈Ω|(G(D)f)(x)=F(x)}, with f∈Wg,1loc(Ω,Ck) is said to be a G-primitivity domain of F. We provide some results about the structure of G-primitivity domains of F at the points of the (suitably defined) G-nonintegrability set of F. A Lusin type theorem for G(D) is also provided. Finally, we give applications to the Maxwell type system and to the multivariate Cauchy-Riemann system.