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

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

2001-09-07

arxiv.org/abs/math/0109056v1

Math. Ann. 320 (2001), 463-485

Mathematics

Complex Variables

20 pages

Scientific paper

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, $u\in C^1(\Omega)$ 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 $\partial\Omega$.

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