An F. and M. Riesz theorem for planar vector fields

Abstract

We prove that solutions of the homogeneous equation $Lu=0$, where $L$ is a locally integrable vector field with smooth coefficients in two variables possess the F. and M. Riesz property. That is, if $Ω$ is an open subset of the plane with smooth boundary, $u\in C^1(Ω)$ satisfies $Lu=0$ on $Ω$, has tempered growth at the boundary, and its weak boundary value is a measure $μ$, then $μ$ is absolutely continuous with respect to Lebesgue measure on the noncharacteristic portion of $\partialΩ$.