We know that the wave operator in $\mathbb{R}^{2}$:

$L=\frac{\partial^2}{\partial x^2}-\frac{\partial^2}{\partial y^2}$

Can I say that $L-\lambda$, $\lambda(x,y) \in C^\infty(\mathbb{R}^2)$ is not hypoelliptic?

I tried use $\lambda$ with rapidly decay (to apply the Fourier transform and find an fundamental solution) to find $\lambda$ such that $L-\lambda$ is hipoelliptic, but it isn't so easy.


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