Continuous solutions for divergence-type equations associated to elliptic systems of complex vector fields

Abstract

In this paper, we characterize all the distributions F∈D′(U) such that there exists a continuous weak solution v∈C(U,Cn) (with U⊂Ω) to the divergence-type equationL1⁎v1+...+Ln⁎vn=F, where {L1,…,Ln} is an elliptic system of linearly independent vector fields with smooth complex coefficients defined on Ω⊂RN. In case where (L1,…,Ln) is the usual gradient field on RN, we recover the classical result for the divergence equation proved by T. De Pauw and W. Pfeffer. Its proof is based on the closed range theorem and inspired by [3] and [6] in the classical case. Our method slightly differs from theirs by relying on the Banach–Grothendieck theorem and introducing tools from pseudodifferential operators, useful in our local setting of a system of complex vector fields with variable coefficients.