## Notes of Nicola Garofalo’s lecture nr 2

1. Stratified nilpotent groups

aka Carnot groups.

1.1. Examples: Heisenberg groups

It was known to physicists under the name Weyl’s group. It was re-christened Heisenberg group by Elias Stein and his school of harmonic analysis.

${\mathbb{H}^n}$ is a multiplication on ${\mathbb{C}^n\times{\mathbb R}}$. The Lie algebra ${\mathfrak{h}^n=\mathbb{C}^n\oplus{\mathbb R}:=V_1\oplus V_2}$ with ${[V_1,V_1]=V_2}$, all other brackets vanishing. The bracket on ${V_1}$ is the symplectic form ${\Im m(z\cdot\bar{z'})}$. Spelling ${z=x+iy}$,

$\displaystyle \begin{array}{rcl} [x+iy,x'+iy']=\frac{1}{2}\sum_{j}x_jy'_j-x'_jy_j. \end{array}$

We assign a formal degree 1 to ${V_1}$ and 2 to ${V_2}$ and define the nonisotropic Lie algebra dilation

$\displaystyle \begin{array}{rcl} \Delta_\lambda(z,t)=(\lambda z,\lambda^2 t). \end{array}$

The nonisotropic group dilation ${\delta_\lambda}$ is given by the same formula. Its Jacobian equals ${\lambda^{2n+2}}$. The exponent ${Q=2n+2}$ is called homogeneous dimension, it will play the role of dimension in Euclidean analysis.

1.2. Carnot groups

A stratified nilpotent Lie algebra of step ${r}$ takes the form ${\mathfrak{g}=V_1\oplus\cdots\oplus V_r}$ with ${[V_1,V_j]=V_{j+1}}$, ${j=1,\ldots,r-1}$, and ${[V_1,V_r]=0}$.

A Lie group ${G}$ is a Carnot group of step ${r}$ if the corresponding Lie algebra ${\mathfrak{g}}$ is stratified nilpotent of step ${r}$.

Define ${\Delta_\lambda(\xi_1+\cdots+\xi_r)=\lambda\xi_1+\cdots+\lambda^r\xi_r}$. These are Lie algebra automorphisms. Since ${G}$ is nilpotent, the group exponential map ${\exp:\mathfrak{g}\rightarrow G}$ is an analytic diffeomorphism. So ${\delta_\lambda=\exp\Delta_\lambda\exp^{-1}}$ are group automorphisms, the nonisotropic group dilations.

Fix an inner product on ${\mathfrak{g}}$ that makes ${V_j}$‘s orthogonal. The nonisotropic gauge on ${\mathfrak{g}}$ is

$\displaystyle \begin{array}{rcl} |\xi|=(\sum\|\xi_s\|^{2r!/s})^{1/2r!}. \end{array}$

It is homogeneous of degree one under dilations ${\Delta_\lambda}$. This is not a norm in the usual sense. Never mind. Using ${\exp}$, we get a nonisotropic gauge on ${G}$, which is homogeneous under ${\delta_\lambda}$.

The Baker-Campbell-Hausdorff formula expresses the multiplication rule of ${G}$ in exponential coordinates,

$\displaystyle \begin{array}{rcl} \exp^{-1}(\exp(\xi)\exp(\eta))=\xi+\eta+\frac{1}{2}[\xi,\eta]+\frac{1}{12}([\xi,[\xi,\eta]]-[\eta,[\xi,\eta]])+\cdots \end{array}$

The full series can be found in books. For nilpotent groups, it is a finite sum. Exercise: compute the multiplication for ${\mathbb{H}^1}$.

1.3. The sub-Laplacian

Let ${e_1,\ldots,e_m}$ be an orthonormal basis of ${V_1}$. They represent left-invariant vectorfields on ${G}$: ${X_i(g)=(L_g)_*(e_j)}$ where ${L_g}$ is left translation by ${g}$, ${L_g(g')=gg'}$. Define

$\displaystyle \begin{array}{rcl} \Delta_{H}=\sum X_j^2. \end{array}$

It depends on the choice of orthonormal basis. Thanks to Hörmander’s theorem, ${\Delta_H}$ is hypoelliptic. It is left-invariant and homogeneous of degree 2 under dilations.

1.4. The ${J}$-map

For step 2 groups, the following map is useful. Assume that the chosen inner product on ${\mathfrak{g}}$ makes ${V_1}$ and ${V_2}$ orthogonal. ${J:V_2\rightarrow End(V_1)}$ is defined by

$\displaystyle \begin{array}{rcl} \langle J(t)z,z'\rangle=\langle[z,z'],t\rangle. \end{array}$

For convenience, fix an orthonormal basis ${\epsilon_1,\ldots,\epsilon_k}$ of ${V_2}$.

Proposition 1 Let ${G}$ be a step 2 Carnot group. Then

$\displaystyle \begin{array}{rcl} X_j&=&\partial_{z_j}+\frac{1}{2}\sum\langle J(\epsilon_\ell),e_j\rangle\partial_{t_\ell},\\ \Delta_H&=&\Delta_z+\frac{1}{4}\sum_{\ell,\ell'}\langle J(\epsilon_\ell)z,J(\epsilon_{\ell'}z\rangle\cdot\partial^{2}_{\ell,\ell'}+\sum_{\ell}\partial_{t_\ell}\sum_i\langle J(\epsilon_\ell)z,e_i\rangle \partial_{z_i}. \end{array}$

1.5. Groups of Métivier type

These are the step 2 Carnot groups for which ${J}$ is nondegenerate,

$\displaystyle \begin{array}{rcl} |J(t)z|\geq\mathrm{const.}\,|z||t|. \end{array}$

1.6. Groups of Heisenberg type

These are the step 2 Carnot groups for which ${J}$ is isometric,

$\displaystyle \begin{array}{rcl} \langle J(t)z,J(t)z'\rangle = |t|^2\langle z,z'\rangle. \end{array}$

They were introduced by Aroldo Kaplan in research concerning hypoellipticity. Although it is a much smaller class that Métivier’s, it turns out to be quite rich. It contains quaternionic Heisenberg groups and many variants. It played a historical role in progress in understanding hypoellipticity.

For such groups, the formula for ${\Delta_H}$ simplifies,

$\displaystyle \begin{array}{rcl} \Delta_H=\Delta_z+\frac{|z|^2}{4}\Delta_t+\sum_{\ell}\sum_i\langle J(\epsilon_\ell)z,e_i\rangle \partial_{z_i}\partial_{t_\ell}. \end{array}$