Some Results About the Structure of Primitivity Domains for Linear Partial Differential Operators with Constant Coefficients

Abstract

AbstractLet G(D) be a linear partial differential operator on $${\mathbb {R}}^n$$ R n , with constant coefficients. Moreover let $$\Omega \subset {\mathbb {R}}^n$$ Ω ⊂ R n be open and $$F\in L^1_{\text {loc}} (\Omega , {\mathbb {C}}^N)$$ F ∈ L loc 1 ( Ω , C N ) . Then any set of the form $$\begin{aligned} A_{f,F}:= \{ x\in \Omega \, \vert \, (G(D)f)(x)=F(x)\}, \text { with }f\in W^{g,1}_{\text {loc}}(\Omega , {\mathbb {C}}^k) \end{aligned}$$ A f , F : = { x ∈ Ω | ( G ( D ) f ) ( x ) = F ( x ) } , with f ∈ W loc g , 1 ( Ω , C k ) 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.