## Lecture 6. Subelliptic diffusion operators

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 $L$ be a diffusion operator with smooth coefficients which is defined on an open set $\Omega \subset \mathbb{R}^n$. We say that $L$ is subelliptic on $\Omega$, if for every compact set $K \subset \Omega$, there exist a constant $C$ and $\varepsilon >0$ such that for every $u \in C^\infty_0(K)$,
$\| u \|^2_{(2\varepsilon)} \le C \left( \| Lu \|_2^2+\|u\|^2_2\right).$

In the above definition, we denoted for $s\in \mathbb{R}$, the Sobolev norm

$\|f \|^2_{(s)}=\int_{\mathbb{R}^n} | \hat{f} (\xi) |^2 (1+\| \xi \|^2)^s d\xi <+\infty ,$

where $\hat{f} (\xi)$ is the Fourier transform of $f$, and $\| \cdot \|_2$ is the classical $L^2$ norm. It is well-known (Weyl’s theorem) that elliptic operators are subelliptic in the sense of the previous definition with $\varepsilon=1$. There are many interesting examples of diffusion operators which are subelliptic but not elliptic. Let, for instance,

$L=\sum_{i=1}^d V_i^2 +V_0$
where $V_0,V_1,\cdots,V_d$ are smooth vector fields defined on an open set $\Omega$. We denote by $\mathfrak{V}$ the Lie algebra generated by the $V_i$‘s, $1 \le i \le d$, and for $x \in \Omega$,

$\mathfrak{V}(x)=\{ V(x), V \in \mathfrak{V} \}.$
The celebrated Hormander’s theorem states that if for every $x \in \Omega$, $\mathfrak{V}(x)=\mathbb{R}^n$, then $L$ is a subelliptic operator. In that case $\varepsilon$ is $1/d$, where $d$ is the maximal length of the brackets that are needed to generate $\mathbb{R}^n$.

If $L$ 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

$\| u \|^2_{(2\varepsilon+s)} \le C \left( \| Lu \|_{(s)}^2+\|u\|^2_{(s)}\right), \quad u \in C^\infty_0(K),$
where $s\in \mathbb{R}$ and the constant $C$ only depends on $K$ and $s$. This implies, in particular, by a usual bootstrap argument and Sobolev lemma that subelliptic operators are hypoelliptic. Iterating the latter inequality also leads to

$\| u \|^2_{(2k\varepsilon)} \le C \sum_{j=0}^k \| L^j u\|^2_2, \quad u \in C^\infty_0(K),$
where $k \ge 0$. This may be used to bound derivatives of $u$ in terms of $L^2$ norms to iterated powers of $u$. Indeed, if $\alpha$ is a multi-index and $k$ is such that $4k\varepsilon> 2 | \alpha | +n$, then we get $\sup_{x \in K} | \partial^\alpha u (x) |^2 \le C\| u \|^2_{(2k\varepsilon)}$ and therefore

$\sup_{x \in K} | \partial^\alpha u (x) |^2 \le C' \sum_{j=0}^k \| L^j u\|^2_2.$
Along the same lines, we also get the following result.

Proposition: Let $L$ be a subelliptic diffusion operator with smooth coefficients on an open set $\Omega \subset \mathbb{R}^n$. Let $u \in L^2(\Omega)$ such that, in the sense of distributions,

$Lu,L^2u,\cdots, L^ku \in L^2(\Omega),$
for some positive integer $k$. Let $K$ be a compact subset of $\Omega$ and denote by $\varepsilon$ the subellipticity constant. If $k>\frac{n}{4 \varepsilon}$, then $u$ is a continuous function on the interior of $K$ and there exists a positive constant $C$ such that

$\sup_{x \in K} | u(x) |^2 \le C \sum_{j=0}^k \|L^j u \|^2_{ L^2(\Omega)}.$
More generally, if $k>\frac{m}{2\varepsilon}+\frac{n}{4\varepsilon}$ for some non negative integer $m$, then $u$ is $m$-times continuously differentiable in the interior of $K$ and there exists a positive constant $C$ such that

$\sup_{|\alpha| \le m} \sup_{x \in K} |\partial^\alpha u(x) |^2 \le C \sum_{j=0}^k \|L^j u \|^2_{ L^2(\Omega)} .$

As a consequence of the previous result, we see in particular that
$\bigcap_{k \ge 0} \mathcal{D}(L^k) \subset C^\infty(\mathbb{R}^n).$

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

Definition: Let $L$ be a diffusion operator on a manifold $\mathbb{M}$. We say that $L$ is subelliptic on $\mathbb{M}$ if it is in any local chart.

The previous Proposition  can then be extended to the manifold case:
Proposition: Let $\mathbb{M}$ be a manifold endowed with a smooth positive measure $\mu$, and let $L$ be a subelliptic diffusion operator with smooth coefficients on an open set $\Omega \subset \mathbb{M}$. Let $u \in L_\mu^2(\Omega)$ such that, in the sense of distributions,

$Lu,L^2u,\cdots, L^ku \in L_\mu^2(\Omega),$
for some positive integer $k$. Let $K$ be a compact subset of $\Omega$. There exists a constant $\varepsilon >0$ such that If $k>\frac{n}{4 \varepsilon}$, then $u$ is a continuous function on the interior of $K$ and there exists a positive constant $C$ such that

$\sup_{x \in K} | u(x) |^2 \le C \sum_{j=0}^k \|L^j u \|^2_{ L_\mu^2(\Omega)} .$
More generally, if $k>\frac{m}{2\varepsilon}+\frac{n}{4\varepsilon}$ for some non negative integer $m$, then $u$ is $m$-times continuously differentiable in the interior of $K$ and there exists a positive constant $C$ such that

$\sup_{|\alpha| \le m} \sup_{x \in K} |\partial^\alpha u(x) |^2 \le C \sum_{j=0}^k \|L^j u \|^2_{ L_\mu^2(\Omega)} .$

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