Abstract
Let G(D) be a linear partial differential operator on {mathbb {R}}^n, with constant coefficients. Moreover let Omega subset {mathbb {R}}^n be open and Fin L^1_{text {loc}} (Omega , {mathbb {C}}^N). Then any set of the form Af,F:={x∈Ω|(G(D)f)(x)=F(x)},withf∈Wlocg,1(Ω,Ck)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} A_{f,F}:= \\{ x\\in \\Omega \\, \\vert \\, (G(D)f)(x)=F(x)\\}, \\text { with }f\\in W^{g,1}_{\\text {loc}}(\\Omega , {\\mathbb {C}}^k) \\end{aligned}$$\\end{document}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.
Highlights
(x1, . . . , xn) be the standard coordinates of Rn and G(D) denote the system [Gjl(D)], where Gjl(D) is the linear partial differential operator with constant coefficients obtained by replacing each ξq in Gjl(ξ1, . . . , ξn) with −i∂/∂xq;
Is said to be a G-primitivity domain of F and the following simple fact holds: If F ∈ Wlmoc,1(Ω, CN ) and there is an open ball B ⊂ Ω such that almost all of B is covered by a G-primitivity domain Af,F (i.e., Ln(B\Af,F ) = 0), with f ∈ Wlgo+c m,1(Ω, Ck), one has S(D)F = 0 a.e. in B for all S ∈ Σm
Under suitable conditions, there are G-primitivity domains of F arbitrarily close in measure to Ω, even if F ∈ Wlmoc,1(Ω, CN ) and Ln(Υm F ) > 0
Summary
Any set of the form Af,F := {x ∈ Ω | (G(D)f )(x) = F (x)}, withf ∈ Wlgo,c1(Ω, Ck)
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