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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