A nonsolvable complex vector field with Hölder coefficients

Abstract:

It is known that the equation \[ \frac {{\partial u}}{{\partial t}} - \alpha (\xi ,t)\frac {{\partial u}}{{\partial \xi }} = f(\xi ,\tau )\] is solvable in a neighborhood of the origin provided Im $\alpha$ does not change sign and $\alpha$ is at least Lipschitz smooth. An example is given where solvability fails although $\alpha$ is of Hölder class $\lambda$ for all $0 < \lambda < 1$. Further, the only solutions to \[ \frac {{\partial u}}{{\partial t}} - \alpha (\xi ,t)\frac {{\partial u}}{{\partial \xi }} = 0\] are the constant functions.