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 $\Omega$ is an open subset of the plane with smooth boundary, $u\in C^1(\Omega)$ satisfiesLu=0 on $\Omega$ , has tempered growth at the boundary, and its weak boundary value is a measure $\mu$ , then $\mu$ is absolutely continuous with respect to Lebesgue measure on the noncharacteristic portion of $\partial\Omega$ .