Theorem 14.22.1.919310C73DBE Let $u,v\in C^2(X)$ satisfy $A,B\geq 0$ and $A^ n=B^ n$, and put $w=u-v$.
If there is a continuous $h>0$ such that $\mathrm{dd}^{\mathrm{c}}(hT)=0$ in distributions, then
24A83D889133
14.22.2
\begin{equation} \mathrm{i}\, \partial w\wedge \bar\partial w\wedge T\equiv 0. \end{equation}
It follows that $w$ is constant if the set where $T$ is strictly positive is dense, or if the reachable set of one point is dense.
Without the weight hypothesis, the sets where $w$ attains its maximum and its minimum are invariant under the flow of every locally Lipschitz vector field with values in $\operatorname {im}Q^\sharp $. If every control orbit is dense, then $w$ is constant. For smooth data it is enough that smooth vector fields with values in $\operatorname {im}Q^\sharp $, for example the columns of the symbol $a$ in (14.22.10) below, together with their iterated Lie brackets, span $TX$ at every point.
of 14.22.1(1).
The product rule gives
A2BD70AF14EC
14.22.5
\begin{equation} \mathrm{dd}^{\mathrm{c}}(w^2)=2w\mathrm{dd}^{\mathrm{c}}w+2\mathrm{d}w\wedge \mathrm d^{\mathrm c}w =2w\mathrm{dd}^{\mathrm{c}}w+\frac{\mathrm{i}}{\pi }\partial w\wedge \bar\partial w. \end{equation}
Approximate $w^2$ in $C^2$ by smooth functions $\varphi $ in (14.22.4). Since $hT$ is continuous and $\mathrm{dd}^{\mathrm{c}}w\wedge T=0$ by (14.20.1), this gives
A832621F5643
14.22.6
\begin{equation} 0=\frac1\pi \int _ X h\, \mathrm{i}\partial w\wedge \bar\partial w\wedge T. \end{equation}
The continuous integrand is nonnegative, so it vanishes identically, which is (14.22.2).
By (14.21.7), (14.22.2) says that $Q(\mathrm{d}w)=0$. For the nonnegative quadratic form $Q$, this implies $Q(\mathrm{d}w,\eta )=0$ for every $\eta $. Therefore
A81FFABB4E8D
14.22.7
\begin{equation} \mathrm{d}w_ x\bigl (Q_ x^\sharp \eta \bigr )=0\qquad (x\in X,\ \eta \in T_ x^*X). \end{equation}
By the chain rule and (14.22.7), $w$ is constant along every absolutely continuous curve whose velocity lies in $\operatorname {im}Q^\sharp $ almost everywhere. Thus $w$ is constant on each reachable set, and by continuity one dense reachable set suffices. If instead $Q$ is positive definite on a dense set, (14.22.6) gives $\mathrm{d}w=0$ there, and hence everywhere.
$\square$
of 14.22.1(2).
Apply the preceding argument to $W=w$ and to $W=-w$ in each chart; both satisfy $LW=0$ by (14.20.1). The maximum set of $w$ is nonempty, closed, and contains every control orbit starting in it. If every orbit is dense, this maximum set is all of $X$, proving constancy. The argument for the minimum set is identical.
It remains to prove the bracket criterion. Suppose that a family of smooth vector fields with values in $\operatorname {im}Q^\sharp $ is bracket generating. At $p$, let $\mathcal D_ p$ be the span of the values of all original fields pushed forward by finite compositions of their local flows and inverse flows. This distribution contains the original fields and is invariant under their flows. Differentiation of pushed-forward fields along a flow shows that it contains their nested Lie brackets; finite-dimensional subspaces are closed, so the derivative remains in $\mathcal D_ p$. Induction and the Jacobi identity give every iterated bracket. The assumed rank condition therefore gives $\mathcal D_ p=T_ pX$.
Choose $2n$ such pushed-forward fields independent at $p$. The flow of each is a conjugate of an original flow and hence preserves its orbit. The map obtained by successively flowing for $2n$ small times has invertible derivative at zero. The inverse function theorem puts a neighborhood of $p$ in its orbit. All orbits are open, and their partition of connected $X$ has only one member. This proves the Hörmander-type sufficient condition; it requires neither a constant-rank hypothesis nor a smooth matrix square root of $a$. In particular, the columns of a smooth symbol $a$ may be used as the original vector fields.
$\square$
Pome
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