This lecture is an introduction to the regularity theory of diffusion operators. Most of the statements will be given without proofs. For a good and easy introduction to the theory in the elliptic case, we refer to the book : Introduction to partial differential equations by Folland. For the proof of Hormander’s theorem, we refer to Hairer’s lecture notes and the references therein.

**Definition:** Let be a diffusion operator with smooth coefficients which is defined on an open set . We say that is subelliptic on , if for every compact set , there exist a constant and such that for every ,

In the above definition, we denoted for , the Sobolev norm

where is the Fourier transform of , and is the classical norm. It is well-known (Weyl’s theorem) that elliptic operators are subelliptic in the sense of the previous definition with . There are many interesting examples of diffusion operators which are subelliptic but not elliptic. Let, for instance,

where are smooth vector fields defined on an open set . We denote by the Lie algebra generated by the ‘s, , and for ,

The celebrated Hormander’s theorem states that if for every , , then is a subelliptic operator. In that case is , where is the maximal length of the brackets that are needed to generate .

If is a subelliptic diffusion operator, using the theory of pseudo-differential operators, it can be proved that the subellipticity defining inequality self-improves into a family of inequalities of the type

where and the constant only depends on and . This implies, in particular, by a usual bootstrap argument and Sobolev lemma that subelliptic operators are hypoelliptic. Iterating the latter inequality also leads to

where . This may be used to bound derivatives of in terms of norms to iterated powers of . Indeed, if is a multi-index and is such that , then we get and therefore

Along the same lines, we also get the following result.

**Proposition: **Let be a subelliptic diffusion operator with smooth coefficients on an open set . Let such that, in the sense of distributions,

for some positive integer . Let be a compact subset of and denote by the subellipticity constant. If , then is a continuous function on the interior of and there exists a positive constant such that

More generally, if for some non negative integer , then is -times continuously differentiable in the interior of and there exists a positive constant such that

As a consequence of the previous result, we see in particular that

We can define subelliptic operators on a manifold by using charts:

**Definition: **Let be a diffusion operator on a manifold . We say that is subelliptic on if it is in any local chart.

The previous Proposition can then be extended to the manifold case:

**Proposition: **Let be a manifold endowed with a smooth positive measure , and let be a subelliptic diffusion operator with smooth coefficients on an open set . Let such that, in the sense of distributions,

for some positive integer . Let be a compact subset of . There exists a constant such that If , then is a continuous function on the interior of and there exists a positive constant such that

More generally, if for some non negative integer , then is -times continuously differentiable in the interior of and there exists a positive constant such that