Ball-Box Theorem and Sequence of Distributions

Question

Let $(e^k,g^k)$ be a sequence of 2d smooth distributions in $R^3$ (with Euclidean metric) s.t $e^k,g^k$ are orthogonal. Let $f^k$ normal direction to this distribution. Suppose $[e^k,g^k] \neq 0 $ on some domain $D(x)$ around x. Moreover let $(e^k,g^k)$ converge to another distribution $(e,g)$

i). The ball-box theorem states that there exists constants $C_k,c_k$ and $r$ s.t

$exp_x(Box(c_kr)) \subset B^k_{sR}(r) \subset exp_x(Box(C_kr))$

where $B^k_{sR}(r)$ is the subriemannian ball of $k^{th}$ distribution. Now my first question is how does these constants depend on the distribution. From the proofs I have studies it seemes upper constant can be chosen to only depend on behaviour of $e^k,g^k$. The lower constant however depends on the map which exponentiates $e^k,g^k,[e^k,g^k]$. Thus it seems to me it would be uniform if for instance $(f^k,[e^k,g^k])$ is uniformly bigger than some constant and decreasing if this quantity goes to zero? However the magn. of $c_k$ depends on the implicit inner workings of inverse function theorem (in the proofs I have seen).

ii). Looking back at the ball box theorem if $(f^k,[e^k,g^k])$ goes to 0 then the boxes (or parallelopipeds) $exp_x(Box(c_kr))$ are becoming increasingly thin in the $f^k$ direction therefore the subRiemannian balls are converging to a 2 dimensional hypersurface? I am not sure how this reasoning could be true since it suggest that also the orbit of this distribution for time less than r also converges to a plane. But even if $(f^k,[e^k,g^k])$ goes to 0, further Lie brackets may be non-zero thus contributing to the orbit. Moreover even the limit distribution e,g may be non-involutive since the Lie brackets might not converge. These are the reasons why this reasoning seems faulty to me.

I am new to sR geometry so far have looked at book/notes of ivanov, agrachev and gromov. I would also be grateful if you have seen and could direct me to any resources where sequence of distributions are handled or studied.

Answer 1

I do not understand exactly what you mean by $\exp_x$, but I guess it is something like the diffeomorphism proposed by Sergei Ivanov. I want to stress out that the ball-box theorem holds only if the coordinates are privileged w.r.t. $(e_k,g_k)$, so you should check that the $exp_x$ you are using defines privileged coordinates for any $k$, in order to apply this theorem.

Regarding the question about $[e_k,g_k]\rightarrow [e,g]\in\text{span}(e,g)$, it is a quite delicate matter. Indeed, this is exactly what happens when we try to get uniform estimates for the sub-Riemannian balls on a sequence of points approaching a singular point (think for example of the Martinet distribution, which is of step 1 outside the plane $\{y=0\}$, where it is step 2, and take any sequence of points $(x_k,y_k,z_k)$ with $y_k>0$ but $y_k\rightarrow 0$). This problem has been treated in the paper

F. Jean, "Uniform Estimation of Sub-Riemannian Balls", Journal on Dynamical and Control Systems, vol. 7 (4), 2001 (http://www.ensta.fr/~fjean/jdcs03_07_01.pdf)

Also, in the notes on s-R geometry by Jean (http://arxiv.org/abs/1209.4387) you can find a somewhat more detailed discussion of privileged coordinates and their subtleties than in the Agrachev, Barilari, Boscain ones.

Answer 2

Dealing with such a low regularity is a tricky business. However in dimension 3 you can get away with your set of assumptions.

First, if the distributions are uniformly Lipschitz and converge in $C^0$, then for every fixed $r$ the $r$-balls converge in the Hausdorff metric. This is easy to see if you choose coordinates $(x,y,z)$ in $\mathbb R^3$ so that the planes of the distributions are separated away from the $z$-axis. In this case every path $t\mapsto(x(t),y(t))$ in the $xy$-plane starting at $(0,0)$ uniquely determines a path $t\mapsto(x(t),y(t),z(t))$ tangent to the distribution and starting at $(0,0,0)$. The coordinate $z(t)$ obtained as the solution of an O.D.E., and these solutions converge since the coefficients are uniformly Lipschitz and converge in $C^0$.

However, the convergence of balls doest not mean the the full orbits converge. If the orthogonal component of the Lie bracket does not vanish, then the orbit is the entire space no matter what. Yet the limit distribution may be integrable and hence have two-dimensional orbits. There is nothing mysterious in that. The orbit is the set of points where the sub-Riemannian distance to the origin is finite. It can be finite for every $k$ but go to infinity as $k\to\infty$, so the limit orbit can be smaller. You can see a similar effect if you consider just the images of linear maps $\mathbb R\to\mathbb R$: for a map $x\mapsto \frac1k x$ the image is the entire line, but the limit map is zero.

As for the "thickness" of balls, they are bounded above in terms of the maximum of $([e,g],f)$ over a neighborhood. And from below in terms of the minimum of $([e,g],f)$. So if the bracket goes to 0 uniformly, you have an expected upper bound for the thickness. But if you want to control the thickness more tightly, you probably need to assume something to prevent fast oscillation of $([e,g],f)$ between zero and the maximum value (or between positive an negative).

You can prove the above mentioned bounds "by hand" if you choose a suitable coordinate system. For example, the one given by a local diffeomorphism $$ (x,y,z) \mapsto \exp(xe)\circ\exp(yg)\circ\exp(zf) . $$ Of course you may want to estimate the distortion of this coordinate system with respect to the original one. This is in some sense "inner working of the inverse function theorem", but you can do it directly by ODE analysis.

In these coordinates, one can move from $(x,y,z)$ to $(0,0,z)$ cheaply: just move time $-x$ along $e$, then time $-y$ along $g$. So you only need to control the changes of the $z$-coordinate as you move along the distribution. The vector field $e$ in these coordinates is the coordinate field $\partial/\partial x$, and $[e,g]=\partial g/\partial x$, so everything is indeed controlled by the 3rd coordinate og $[e,g]$.

Answer 3

Even though this question already has a great answer, I think there's more to say.

Let's rephrase the question informally as "what do the constants in the ball-box theorem (and its generalizations) depend on?" After all, the original question wants constants which are uniform over a sequence, so if we can make explicit the dependence of our constants in the right way, perhaps we can show that it is uniform over an appropriate sequence. I'll answer this more vague question to the best of my ability.

As the question points out, the original proofs of ball-box type theorems rested on some kind of implicit function theorem--and this step greatly influenced the estimates. As I will discuss below, there are applications where this is not good enough. Fortunately, Tao and Wright gave an alternative approach to these results which deftly side-steps this and other problems and paves the way to stronger and more general results. From a technical standpoint, their approach opens the door to improving the dependence of constants in results which generalize the ball-box theorem. Results which are rooted in Tao and Wright's approach are not usually explicitly tied back to the original ball-box theorem (for reasons I'll discuss), and so it can be hard for someone starting out to connect all these results. I hope this post may help.

In what follows, I'll often refer to two important papers on the subject:

• [NSW] Nagel, Stein, and Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985), no. 1-2, 103–147. MR 793239
• [TW] Tao and Wright, $L^p$ improving bounds for averages along curves, J. Amer. Math. Soc. 16 (2003), no. 3, 605–638.

To begin with, let's discuss some of [NSW] and in what ways it is not sufficient for some applications. Then we'll turn to results rooted in [TW], which are in many ways optimal.

Let $V_1,\ldots, V_r$ be smooth vector fields on a manifold $M$. Define the sub-Riemannian ball of radius $1$, centered at $x\in M$, in terms of $V=V_1,\ldots, V_r$ by:

$$B_V(x,1):=\left\{ y\in M : \exists \gamma:[0,1]\rightarrow M, \gamma(0)=x, \gamma(1)=y, \gamma'(t)=\sum_{j=1}^r a_j(t) V_j(\gamma(t)), \left\| \sum_{j} |a_j|^2\right\|_{L^\infty}<1\right\}.$$

Set $$B_V(x,\delta):=B_{\delta V_1,\ldots, \delta V_r}(x,1).$$

Let $W_1,\ldots, W_r$ be smooth vector fields on a manifold $M$ of dimension $n$, satisfying H\"ormander's condition of order $m$. To each $W_j$ assign the formal degree $1$. If $V$ has formal degree $e$, we assign to $[W_j, V]$ the formal degree $e+1$. Let $(X_1,d_1),\ldots, (X_q,d_q)$ be an enumeration of all such vector fields of formal degree $\leq m$. By the assumption of H\"ormander's condition, $X_1,\ldots, X_q$ span the tangent space to $M$ at every point. We let $(X,d)$ denote the list $(X_1,d_1),\ldots, (X_q,d_q)$. A key property of these vector fields, which we will come back to, is:

$$[X_i,X_j]=\sum_{d_k\leq d_i+d_j} c_{i,j}^k X_k, \quad c_{i,j}^k\in C^\infty(M),\quad (1)$$ indeed, if $d_i+d_j\leq m$, this follows from the Jacobi identity, while if $d_i+d_j>m$ this follows from the fact that $X_1,\ldots, X_q$ span the tangent space.

We set

$$B_{(X,d)}(x,\delta):=B_{\delta^{d_1}X_1,\ldots, \delta^{d_q}X_q}(x,1).$$ [NSW] showed the balls $B_{(X,d)}(x,\delta)$ and $B_W(x,\delta)$ were comparable, so let's use the former instead.

Given $x\in M$, $\delta>0$, pick $j_1=j_1(x,\delta),\ldots, j_n=j_n(x,\delta)$ so that

$$\left| \det \left( \delta^{d_{j_1}} X_{j_1}(x) |\cdots| \delta^{d_{j_n}} X_{j_n}(x)\right)\right|$$ is maximal among all such choices of $j_1,\ldots, j_n$. Set

$$\Psi_{x,\delta}(t_1,\ldots, t_n):= \exp\left( t_1 X_{j_1}+\cdots+t_n X_{j_n}\right)x.$$

The Ball-Box Theorem [NSW] (informally): $B_{(X,d)}(x,\delta)$ is ``comparable'' to the box $$\Psi_{x,\delta}( [-\delta^{d_{j_1}}, \delta^{d_{j_1}}] \times \cdots\times [-\delta^{d_{j_n}},\delta^{d_{j_n}}])$$

Instead of trying to make $B_{(X,d)}(x,\delta)$ look like a small rectangle, it is often more convienient to make it look like a ball of essentially unit radius. So let's set $$\Phi_{x,\delta}(t_1,\ldots, t_n):= \Psi_{x,\delta}(\delta^{d_{j_1}}t_1,\ldots, \delta^{d_{j_n}} t_n).$$

Another Ball-Box-type Theorem [NSW]: There exists $\eta,\xi\approx 1$ such that $$B_{(X,d)}(x,\xi\delta) \subseteq \Phi_{x,\delta}(B^n(\eta))\subseteq B_{(X,d)}(x,\delta). $$ Futhermore, the vector fields $\Phi_{x,\delta}^{*} \delta_{d_j} X_j$ are $C^\infty$ and span the tangent space uniformly in $\delta$. I.e., the vector fields $\Phi_{x,\delta}^{*} \delta W_1,\ldots, \Phi_{x,\delta}^{*} \delta W_r$ satisfy H\"ormander's condition uniformly in $\delta$. In short, the map $\Phi_{x,\delta}$ ``rescales'' the case of $\delta$ small to the case $\delta=1$. The implicit constants are independent of $x\in M$ and $\delta\in (0,\delta_0]$ where $\delta_0>0$ is some small number.

It will be more convienient in a moment to work on the Euclidean unit ball $B^n(1)$ instead of $B^n(\eta)$, and this can be achieved by replacing $\Phi_{x,\delta}(t)$ with $\Phi_{x,\delta}(\eta t)$. In this incarnation, it says that the ball-box theorem means the sub-Riemannian ball $B_{(X,d)}(x,\delta)$ ``looks like'' the Euclidean ball $B^n(1)$ in an appropriate coordinate system in a nice way. This is why we don't usually refer to generalizations of this result as ball-box theorems: they're more like ball-ball theorems, though that isn't very descriptive.

There are two ways that the above theorems are not quite good enough for some applications in harmonic analysis:

1. Above, we discussed balls of the form $B_{\delta W_1,\ldots, \delta W_r}(x,1)$. But what if we instead wanted to study $B_{\delta_1 W_1,\ldots, \delta_r W_r}(x,1)$ and obtain results which are uniform for $\delta_1,\ldots, \delta_r$ small? What assumptions on $W_1,\ldots, W_r$ allow us to obtain uniform estimates? This so-called multi-parameter situaiton arose in some questions and is easily seen to beyond the methods of [NSW]. The methods of [NSW] are good enough if $\delta_1,\ldots, \delta_r$ are assumed to be weakly-comparable (i.e., $\delta_j^N\lesssim \kappa \delta_k$ for all $j,k$), but not if the $\delta$s are unrestricted. One could also create other situations by making the vector fields depend on $\delta$ in a complicated way, and it is desirable to study these as well.
2. The conclusions of the above ball-box-type theorems are clearly invariant under arbitrary $C^2$ diffeomorphisms. But the methods of [NSW] are not: they rely on estimates of the $C^m$ norms of the coeffients of the vector filds in some fixed coordinate system.

In Section 4 of [TW], Tao and Wright sketched another approach to the main results of [NSW]; despite the fact that [TW] was working in the weakly comparable setting and they acknoledged that the methods of [NSW] were sufficient to prove what they needed. Perhaps for this reason, these methods were described without much explantion for why they were better. It took more than 10 years before it was explicitly pointed out that their methods avoided the two above issues.

To describe what is possible using their methods, a change in perspective is useful. Set $$X^\delta_j := \delta^{d_j} X_j,$$ $$c_{i,j}^{k,\delta}:= \begin{cases}\delta^{d_i+d_j-d_k} c_{i,j}^k & d_k\leq d_i+d_j\\ 0&\text{otherwise.} \end{cases}$$

Multiplying both sides of (1) by $\delta^{d_i+d_j}$ we have $$[X_i^{\delta}, X_j^{\delta}] = \sum_k c_{i,j}^{k,\delta} X_k^{\delta},$$ where $c_{i,j}^{k,\delta}$ is $C^\infty$ uniformly in $\delta$.

So here is our new perspective. Let $Z_1,\ldots, Z_q$ be $C^1$ vector fields on a $C^2$ manifold $M$ of dimension $n$, which span the tangent space at every point. Suppose $$[Z_i,Z_j]=\sum_{k} c_{i,j}^k Z_k.$$ We will make sure our results only depend on quantities like: $$\sup_{x} \sum_{|\alpha|\leq l} |Z^{\alpha} c_{i,j}^k(x)|,\quad (2)$$ for some $l$, where $\alpha$ ranges over ordered multi-indices (since $Z_1,\ldots, Z_q$ may not commute). Thus, whatever results we prove for $Z_1,\ldots, Z_q$ will apply to $Z_1= X_1^{\delta},\ldots, Z_q=X_q^{\delta}$ uniformly in $\delta$. So we can ignore the parameter $\delta$ entirely now. Also, (2) is a quantity which is invariant under arbitrary $C^2$ diffeomorphisms, so if we only use quantities like this and other diffeomorphic invariant quantities, our results will be invariant under arbitrary $C^2$ diffeomorphisms.

Theorem: Fix a point $x_0\in M$ and take $Z_1,\ldots, Z_q$ as above. All implicit constants which follow will be as desrcibed above. There exists a map $\Phi:B^n(1)\rightarrow B_Z(x_0,1)$ which is a $C^2$ diffeomorphism onto its image which is an open subset of $B_Z(x_0,1)$. There exists $\xi\approx 1$ such that $$B_Z(x_0,\xi)\subseteq \Phi(B^n(1))\subseteq B_{Z}(x_0,1)$$ Furthermore, $$\| \Phi^{*} Z_j\|_{C^m}\lesssim 1, \forall m,$$ i.e., $\Phi^{*}Z_1,\ldots, \Phi^{*} Z_q$ are smooth ``uniformly''. These vector fields also span the tangent space ''uniformly'' $$\sup_{j_1,\ldots, j_n} \left| \det\left( \Phi^{*} Z_{j_1} | \cdots | \Phi^{*} Z_{j_n} \right)\right|\approx 1.$$

When applied to the vector fields $\delta^{d_1}X_1,\ldots, \delta^{d_q}X_q$, this yeilds the ball-box type theorem discussed above. But it can also be applied to the multi-parameter setting. Or in the setting of a sequence of vector fields as in the original question. If the hypotheses of the Theorem hold uniformly, then the conclusions do as well. It is also invariant under arbitary $C^2$ diffeomorphisms.

I wasn't explicit above exactly what all the constants depend on. This can be found in this paper. This might be the kind of paper the original question asked for--its main theorem can be applied to a sequence of distributions to sometimes give uniform results.

In this theorem, one can try to use the map $\Phi_{x,\delta}$ for $\Phi$. This works to a certain extent (see this paper, joint with Stovall). However, for sharper results, a different $\Phi$ is used.

It can be quite useful to have the flexibility to use a $\Phi$ other than $\Phi_{x,\delta}$. As just mentioned, it sometimes allows for sharper results. Also, one can ask the same sort of generalized "ball box theorems" in other cateogries. For example, one could want a result on complex manifolds using holomorphic maps. This is also possible--see this paper. It is better to think of these as "scaling" theorems than "ball box" theorems, though, I think.