An F. and M. Riesz theorem for planar vector fields (Q5947140)

The author proves that the solutions of the homogeneous equation \(Lu=0\), where \(L\) is a locally integrable vector field with smooth coefficients in two variables, posses the F. and M. Riesz property. That is, if \(\Omega\) is an open subset of the plane with smooth boundary, \(u\in C^1 (\Omega)\) satisfies \(Lu=0\) 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 \(\partial \Omega\).