Lecture 6. Subelliptic diffusion operators

Posted on September 27, 2016 by Fabrice Baudoin

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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