Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.