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}

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