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Consider a smooth real vector field $X(x,\partial _x)=\sum a_i(x) \frac{\partial}{\partial x_i}$ on an open subset $\Omega \subset \mathbb{R}^n$ (which is assumed to be sufficiently regular) as a linear first-order differential operator. The coefficients $a_i(x)$ are smooth real valued functions.

I have studied the theory of fundamental solutions (in the context of distribution theory) for constant-coefficient linear differential operators (e.g. Laplacian, Heat, Wave...). In general, a vector field has not constant coefficients; anyway it seems to be a quite elementary differential operator, since it's first-order. I'd like to know if something is known about the fundamental solution of a vector field, for example if there is some condition for its existence, or if something can be said about its properties, especially about the hypoellipticity of the vector field as a differential operator.

Since I couldn't find anything on wikipedia, I would also appreciate any reference or anything were I can study these things. Thank you.

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