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Consider the following operator $$\partial_t^2-\partial_x^2+\lambda(x,t)$$ where $\lambda$ is a $C^{\infty}$ function on $\mathbb{R}^2$ and the operators above are second partial derivatives. How can we say it is not hypoelliptic? Well I know that without the term of order zero $\lambda$ it is the wave operator so it not hypoelliptic because a solution of the homogeneous equation is of the form $f(x+t)+g(x-t)$ and so if we take these function not smooth we have the result. But i can't find a solution for this operator so how can i proceed? Thanks

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