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

We prove that solutions of the homogeneous equation Lu = 0, where L is a lo cally integrable vector field with smooth coefficients in two variables pos sess the F. and M. Riesz property. That is, if Omega is an open subset of t he plane with smooth boundary, u is an element of C-1 (Omega) satisfies Lit = 0 on Omega, has tempered growth at the boundary, and its weak boundary v alue is a measure u, then mu is absolutely continuous with respect to Lebes gue measure on the noncharacteristic portion of partial derivative Omega.