## Notes of Nicola Garofalo’s lecture nr 1

Hypoelliptic operators and analysis on Carnot-Carathéodory spaces

1. Hypoelliptic operators

1.1. Motivation: semi-flexible polymers

In 1995, when studying Euler’s elastica, introduced the following differential operator

$\displaystyle M=\partial_\theta^2+\cos\theta \partial_x +\sin\theta \partial_y -\partial_t.$

It turns out to play a role in models of semi-flexible polymers.

Write ${X_1=\partial_\theta}$ and ${X_0=\cos\theta \partial_x +\sin\theta \partial_y -\partial_t}$. Then ${X_2:=[X_1,X_0]=-\sin\theta \partial_x +\cos\theta \partial_y}$, ${X_3:=[X_1,X_2]=-\cos\theta \partial_x -\sin\theta \partial_y}$. Thus ${X_1}$ and ${X_0}$ are bracket generating.

1.2. Hypoellipticity

A differential operator ${P}$ with smooth coefficients is hypoelliptic if ${Pu=f}$ with ${f}$ smooth and ${u}$ a distribution implies that ${u}$ is smooth.

The main example is the Laplacian ${\Delta}$ (sometimes known as Weyl’s Lemma, due to Cacciopoli in 1938, generalized to variable coefficients by Cimino in 1940).

The next example is the heat operator ${\Delta-\frac{\partial}{\partial t}}$. Note that it is not ${C^\omega}$-hypoelliptic. On the other hand, the wave equation ${\Delta-\frac{\partial^2}{\partial t^2}}$ is not hypoelliptic.

Theorem 1 (Hörmander 1967) If ${X_0,X_1,\ldots,X_m}$ are smooth bracket generating vectorfields and ${c}$ is a smooth function, then

$\displaystyle \begin{array}{rcl} L=\sum X_i^2 +X_0+c \end{array}$

is hypoelliptic.

The bracket generating condition is nearly necessary, as shown by Hörmander in his PhD in 1954 (under Gårding).

Note that higher order differential operators are not hypoelliptic.

1.3. Back to Mumford’s operator

Write ${z=x+iy}$ and introduce the group law

$\displaystyle \begin{array}{rcl} (\theta,z,t)(\theta',z',t')=(\theta+\theta),z+e^{i\theta}z',t+t'). \end{array}$

This is the Lie group ${RT\times{\mathbb R}}$, where ${RT}$ denotes the planar roto-translation group. Then Mumford’s operator is left-invariant. In fact, each ${X_i}$ is left-invariant. Note that ${RT}$ is not nilpotent, this group does not have dilations.

1.4. Kolmogorov’s operator

In 1934, in his approach to the kinetic theory of gases, Kolmogorov introduces the equation

$\displaystyle \begin{array}{rcl} Ku=\partial_x^2 +x\partial_y -\partial_t=X_1^2+X_0 \end{array}$

where ${X_1=\partial_x}$ and ${X_0=x\partial_y -\partial_t}$, ${X_2:=[X_1,X_0]=\partial_y}$ generate ${{\mathbb R}^3}$. So Kolmogorov’s operator is hypoelliptic (this conclusion is one of Hörmander’s main motivations). In fact, Kolmogorov’s had computed an explicit fundamental solution for ${K}$, which is smooth outside the diagonal, this implies hypoellipticity.

1.5. Stein’s program

Probabilists, starting from Mark Kac, realized very early that ${M}$ and ${K}$ are related in the same way as a manifold is connected to its tangent space.

In his ICM 1970 address, Stein launched a program of developping noncommutative harmonic analysis by approximating operators by their second order Taylor expansions.

Expanding ${\cos}$ and ${\sin}$ at second order, we approximate ${M}$ with ${L=\partial_\theta^2+\frac{\theta^2}{2}\partial_x+\theta\partial_y-\partial_t}$, which is again hypoelliptic, by Hörmander’s theorem. Switch notation to make ${L=X_1^2+X_0}$ where ${X_1=\partial_x}$, ${X_0=\frac{x^2}{2}\partial_y+x\partial_z-\partial_t}$. Define

$\displaystyle \begin{array}{rcl} \delta_\lambda(x,y,z,t)=(\lambda x,\lambda^4 y,\lambda^3 z,\lambda^2 t). \end{array}$

Then ${(\delta_\lambda)_*L=\lambda^2 L}$.

1.6. Exponential map and group law for ${L=X_1^2+X_0}$

A theorem of Lanconelli states that there is a Lie group underlying every real analytic differential operator admitting dilations, under some bracket generating condition. We perform the calculation for ${L}$.

$\displaystyle \begin{array}{rcl} Exp(uY)(g)=\begin{pmatrix} x+u_1\\y+\frac{u_2}{2}(x^2+\frac{u_1^2}{3}+u_1 x)+u_3 x +\frac{u_1 u_3}{2}+u_4\\z+u_3+u_2 x+\frac{u_1 u_2}{2}\\t-u_2 \end{pmatrix} \end{array}$

From the exponential map, one extracts the group law in ${{\mathbb R}^4}$,

$\displaystyle \begin{array}{rcl} \begin{pmatrix} x\\y\\z\\t \end{pmatrix}\begin{pmatrix} x'\\y'\\z'\\t' \end{pmatrix}=\begin{pmatrix} x+x'\\y+y'+xz'-\frac{t'x^2}{2}\\z+z'-t'x\\t+t' \end{pmatrix}. \end{array}$

The ${X_i}$‘s become left-invariant.

Kolmogorov’s operator differs from ${L}$, the associated Lie group is different. A third operator, similar to ${K}$ and ${L}$, has been studied recently by Citti, Menozzi and Polidoro.

2. Stratified nilpotent Lie groups

Also known as Carnot groups.