Applications of Rademacher's Theorem

Question

Rademacher's Theorem (that every Lipschitz function on $\mathbb{R}^{n}$ is almost everywhere differentiable) is a remarkable result on the structure of the space of Lipschitz functions, but I was wondering whether it has any interesting applications. All of the "useful" results (or maybe "applicable") that I know of about weak versions of differentiability involve estimates (e.g. Sobolev embedding, Lebesgue differentiation theorem).

Answer 1

I will mention seven different applications:

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• Characterization of almost everywhere differentiability.

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The following result is a consequence of the Rademacher theorem:

Theorem (Stepanov). A function $f:\Omega\to\mathbb{R}$ defined on an open set $\Omega\subset\mathbb{R}^n$ is differentiable almost everywhere if and only if $$ \limsup_{y\to x}\frac{|f(y)-f(x)|}{|y-x|}<\infty $$ for almost all $x\in\Omega$.

A beautiful proof of this classical result is given in [4].

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• Uniqueness of the closest point.

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Let $K\subset\mathbb{R}^n$ be a compact set. For $x\in \mathbb{R}^n\setminus K$ let $$ D_x=\{y\in K:\, d(x,y)=\operatorname{dist}(x,K)\}. $$ If the set $D_x$ consists of one point, then it means there there is a unique point in $K$ that is closes to $x$. Unfortunately there might be points where the closest point is not unique i.e. when $D_x$ contains more than one point. For example if $K$ is a sphere centered at $x$, then $D_x=K$. However, the set of non-uniqueness points is small:

Theorem. The set of points $x\in \mathbb{R}^n\setminus K$ such that the closest point in $K$ to $x$ is not unique (i.e. $D_x$ has more than one point) has measure zero.

Proof. Indeed, the distance function is Lipschitz and hence differentiable almost everywhere (by Rademacher). However, if the distance is differentiable at $x$, then $D_x$ consists of one point. For a proof of this fact see https://mathoverflow.net/a/299066/121665. $\Box$

For a related post see: Set of points with a unique closest point in a compact set.

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• Aleksandrov differentiability theorem.

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Aleksandrov proved the following important result about the second order differentiability of convex functions.

Theorem (Aleksandrov). Let $f:\mathbb{R}^n\to\mathbb{R}$ be a convex function. Then $f$ is locally Lipschitz and hence differentiable a.e. (Rademacher). Let $E\subset\mathbb{R}^n$ be the set of points where $f$ is differentiable. Then for almost all $x\in E$ there is a symmetric matrix $D^2f(x)$ such that $$ (1)\ \ \ \ \ \ \ \lim_{y\to x} \frac{|f(y)-f(x)-Df(x)(y-x)-\frac{1}{2}(y-x)^TD^2f(x)(y-x)|}{|y-x|^2}=0 $$ and $$ (2)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \lim_{E\ni y\to x} \frac{|Df(y)-Df(x)-D^2f(x)(y-x)|}{|y-x|}=0. $$

This result can be obtained from the Rademacher theorem. The idea is as follows. The subdifferential $\partial f(x)$ is the set of all $v\in\mathbb{R}^n$ such that $$ f(y)\geq f(x)+\langle v,y-x\rangle \ \quad \text{for all $y\in\mathbb{R}^n$.} $$ It is easy to prove that $\partial f(x)\neq\emptyset$ for all $x$. Let $$ \Gamma f=\{(x,y):\, x\in\mathbb{R}^n,\ y\in \partial f(x)\} $$ be the graph of $x\mapsto\partial f(x)$. Note that this is a multi-valued function since $\partial f(x)$ may have more than one point. However, if you apply a suitable rotation of $\Gamma f$ by $45^o$, $\Gamma f$ will became a graph of a $1$-Lipschitz function $g:\mathbb{R}^n\to\mathbb{R}^n$. Clearly $g$ is differnetiable a.e. so the graph $\Gamma f$ has tangent space almost everywhere (as isometric to the graph of $g$). Since $\Gamma f$ is more or less the graph of the gradient $Df$, it follows that $Df$ is differentiable a.e. as stated in (2). This however, implies (1) too.

For more details, see

[AA] L. Ambrosio, G. Alberti, A geometrical approach to monotone functions in $\mathbb{R}^n$. Math Z. 230(1999), 259-316, DOI: 10.1007/PL00004691.

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• Non-embedding of the Heisenberg group.

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Theorem (Semmes). The Heisenberg group $\mathbb{H}^n$ does not admit a bi-Lipschitz embedding not any Euclidean space.

That was observed by Semmes and it follows from a version of the Rademacher functions on the Heisenberg group proved by Pansu, see [1] page 397. See also Non embedding of the Heisenberg group

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• Kirchheim-Rademacher theorem.

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Suppose that $f:\Omega\to\mathbb{R}^m$ (not necessarily Lipschitz) is differentiable at $x\in\Omega$. Then $$ \left|\frac{|f(y)-f(x)|-|Df(x)(y-x)|}{|y-x|}\right| \leq \frac{|f(y)-f(x)-Df(x)(y-x)|}{|y-x|} \stackrel{y\to x}{\longrightarrow} 0. $$ Observe that $\Vert z\Vert_x:=|Df(x)z|$ is a seminorm (that mens $\Vert z_1+z_2\Vert_x\leq\Vert z_1\Vert_x+\Vert z_2\Vert_x$, $\Vert tz\Vert_x=|t|\Vert z\Vert_x$, but $\Vert\cdot\Vert_x$ may vanish on a subspace of $\mathbb{R}^n$).

Theorem (Kirchheim). If $f:\mathbb{R}^n\supset\Omega\to X$ is a Lipschitz continuous mapping into any metric space $(X,d)$, then for almost all $x\in\Omega$, there is a seminorm $\Vert\cdot\Vert_x$ in $\mathbb{R}^n$ such that $$ \frac{d(f(y),f(x))-\Vert y-x\Vert_x}{|y-x|} \to 0 \quad \mbox{as $y\to x$.} $$

The seminorm $\Vert\cdot\Vert_x$ is called the metric derivative of $f$.

This result was proved in [2]. For another proof see [1].

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• Topological dimension

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The following result can be proved as an application of the Kirchheim-Rademacher theorem along with the Kirchheim area formula.

Theorem. Suppose that $f:\mathbb{R}^n\supset\Omega\to X$, $\Omega$ open, is a Lipschitz continuous map onto a metric space $X$, $f(\Omega)=X$. Then $\operatorname{dim} X=n$ if and only if $\mathcal{H}^n(X)>0$.

For further comments, see Topological dimension, Hausdorff dimension, and Lipschitz mappings.

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• Length preserving mappings

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(See also the answer of Anton Petrunin). In [5], Theorem 2.4.11, Gromov proved that any Riemannian manifold of dimension $n$ admits a mapping into $\mathbb{R}^n$ that preserves lengths of curves. Such a mapping is necessarily Lipschitz and hence differentiable almost everywhere (by Rademacher). One can prove that the Jacobian of such a mapping is different than zero almost everywhere, and hence there is no curve-length preserving mapping into $\mathbb{R}^m$ for $m < n$. More precisely we have.

Theorem. Let $M^n$ and $N^m$ be Riemannian manifolds of dimensions $n$ and $m$ respectively. If $f:M\to N$ is a mapping that preserves length of curves, then it is Lipschitz, differentiable a.e. and $\operatorname{rank} Df= n$ a.e. In particular, we necessarily have $m\geq n$.

Proof. The fact that $f$ is Lipschitz follows from the definition of the Riemannian metric. Hence it is locally Lipschitz when represented in a local coordinate system. The map represented in a coordinate system will not necessarily preserve length, but locally it will satisfy $\ell(f\circ\gamma)\geq C\ell(\gamma)$ and it easily follows that the directional derivatives will satisfy $D_vf\geq C$ a.e. so the derivative is non-degenerate a.e. so its rank is equal $n$.

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References

[1] L. Ambrosio, B. Kirchheim, Bernd Rectifiable sets in metric and Banach spaces. Math. Ann. 318 (2000), 527–555.

[2] B. Kirchheim, Rectifiable metric spaces: local structure and regularity of the Hausdorff measure. Proc. Amer. Math. Soc. 121 (1994), 113-123.

[3] S. Semmes, On the nonexistence of bi-Lipschitz parameterizations and geometric problems about $A_\infty$-weights. Rev. Mat. Iberoamericana 12 (1996), 337-410.

[4] J. Maly, A simple proof of the Stepanov theorem on differentiability almost everywhere. Exposition. Math. 17 (1999), no. 1, 59–61.

[5] M. Gromov, Partial differential relations. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 9. Springer-Verlag, Berlin, 1986.

Answer 2

The one-dimensional Rademacher differentiation theorem implies that the Cartesian product of two compact measure zero subsets of the real line is purely unrectifiable, which in turn can be used to establish the Besicovitch projection theorem, which asserts that if a subset of the plane has finite 1-dimensional Hausdorff measure and is purely unrectifiable, then almost every projection of that set to the real line has measure zero. Thus, for instance, if one takes the Cartesian product $C \times C$ of two copies of the middle-halves Cantor set $\{0,1\} \in C = \frac{1}{4} C \cup (\frac{1}{4} C + \frac{3}{4}) \subset [0,1]$, then almost every line in the plane will fail to hit this set. (Quantitative versions of this problem (commonly known as "Buffon's needle problem") have attracted attention in recent years, see e.g. the recent survey of Laba at https://arxiv.org/abs/1212.0247 .)

A few years ago with Hans Lindblad in https://arxiv.org/abs/1011.0949 , we used the one-dimensional Rademacher differentiation theorem to establish that solutions to a certain nonlinear wave equation in one spatial dimension necessarily decayed to zero as time went to infinity. This is in contrast to the linear wave equation which does not decay in one spatial dimension. The rough idea was that if the solution did not decay, then one could show that it concentrated along a Lipschitz curve in spacetime, which by Rademacher was approximately linear at some locations and some scales, and this could be shown to be in contradiction to a certain Morawetz-type estimate on solutions to nonlinear wave equations.

One fairly well known application of the higher-dimensional Rademacher differentiation theorem is by Pansu who extended this theorem to Carnot groups, and a variant of his theory establishes the fact that if two finitely generated nilpotent groups are quasiisometric, then their associated Carnot groups are isomorphic, which is still one of the strongest statements known about quasiisometry of groups in the nilpotent case.

Answer 3

A Banach space is said to have the RNP provided every Lipschitz function from the line into the space has a point of differentiability. Reflexive spaces and separable dual spaces have the RNP. If $X$ is a separable Banach space and $Y$ has the RNP, Rademacher's theorem is used to proved that every Lipschitz function from $X$ into $Y$ is differentiable off a null set (where null set can have various meanings). A huge number of results in nonlinear geometric functional analysis depend on this. See the book "Geometric nonlinear functional analysis" by Benyamini and Lindenstrauss for some of them.

Answer 4

• There is no path isometry $\mathbb R^2\to\mathbb R$ (search for length-preserving map in my collection);
• There is no path isometry $(\mathbb R^2,\ell_p)\to\mathbb R^n$ for $p\not=2$.

Answer 5

The application I am most familiar with is that it is used in the proof of the following result:

Suppose $f : \mathbb{R}^n \to \mathbb{R}$ is Lipschitz. For any $\epsilon > 0$, there exists a $C^1$ function $g$ such that the Lebesgue measure of the set { $f\neq g$ } $\cup$ { $D f \neq D g $ } is at most $\epsilon$.

One reason I know for this result being useful is that it gives the `approximate' tangent bundle structure on a countably rectifiable set:

A countably $n$-rectifiable subset of Euclidean space is usually defined as a set (almost all of) which is contained in a countable union of Lipschitz images of $\mathbb{R}^n$. The preceeding proposition is used to show that one can replace " Lipschitz images of $\mathbb{R}^n$ " with "embedded $n$-dimensional $C^1$ submanifolds" in this definition.

Since $C^1$ submanifolds have tangent spaces, one is now only a couple of checks away from the fact that one can define the intrinsic derivative of a locally Lipschitz function at almost every point of a countably rectifiable set.

Having this kind of differentiable structure for objects and functions so weak is essential for studying various GMT-esque regularity problems e.g. understanding the singularities of stationary varifolds.

Answer 6

The following application may not be too far outside the realm of usual applications, as described in the OP's original question statement, but I will let others decide for themselves.

Rademacher's theorem is used to prove a maximum/minimum principle for 2D drift-diffusion equations $$\partial_t\theta + u\cdot\nabla\theta + \kappa(-\Delta)^{\gamma/2}\theta, \tag{1}$$ where $\theta:\mathbb{R}^2\rightarrow\mathbb{R}$ is a scalar-valued function, $u:\mathbb{R}^2\rightarrow\mathbb{R}^2$ is a divergence-free vector field possibly depending on $\theta$, $\kappa\geq 0$, and $(-\Delta)^{\gamma/2}$ is the fractional Laplacian of order $\gamma$. In the case where the velocity field $u$ is recovered from $\theta$ through the Biot-Savart law $$u = (-R_2\theta,R_1\theta),$$ where $R_1,R_2$ are the usual 2D Riesz transforms, we recover the well-known dissipative Surface Quasi-Geostrophic (SQG) equation.

When $\kappa=0$, it is easy to see from the existence of a flow map that the minimum and maximum of a smooth solution $\theta$ are conserved. For $\kappa>0$, this is no longer necessarily the case; however, the minimum is nondecreasing and the maximum is nonincreasing. To see this, one argue as follows, as for instance shown in this paper by the Cordobas. Denote the supremum $\theta$ at time $t$ respectively by $$M(t) := \sup_{x\in\mathbb{R}^2} \theta(t,x).$$ If $\theta(0)\in L^\infty$, then since $\|\theta(t)\|_{L^p} \leq \|\theta(0)\|_{L^p}$, for any $1\leq p\leq \infty$, (see the aforementioned paper), it follows that $M(t)$ is finite. Moreover, if $\theta(t)\in \hat{L}^1$ for each $t$, then it follows from the Riemann-Lebesgue lemma that $\theta(t)\in C_0$. Hence, for each $t$, there exists a point $x(t)\in\mathbb{R}^2$ such that $$M(t) = \theta(t,x(t)).$$ It's not hard to show that $M(t)$ is a Lipschitz continuous function. Hence by Rademacher's theorem, $M(t)$ is differentiable for almost every $t$. At such a point $t_0$ of differentiability, we can find by a compactness argument a sequence of positive numbers $h_j\rightarrow 0$ such that $x(t_0+h_j)\rightarrow x_0\in\mathbb{R}^2$ as $j\rightarrow\infty$. Since $M$ is continuous, we then have $M(t_0) = \theta(t_0,x_0)$. Now \begin{equation*} \begin{split} &\frac{M(t_0+h_j)-M(t_0)}{h_j} = \frac{\theta(t_0+h_j,x(t_0+h_j))-\theta(t_0,x_0)}{h_j} \\ &=\frac{\theta(t_0+h_j,x(t_0+h_j))-\theta(t_0,x(t_0+h_j))}{h_j} + \frac{\theta(t_0,x(t_0+h_j))-\theta(t_0,x_0)}{h_j} \\ &\leq \frac{\theta(t_0+h_j,x(t_0+h_j))-\theta(t_0,x(t_0+h_j))}{h_j}, \end{split} \end{equation*} since the second term in the penultimate line is nonpositive. Letting $j\rightarrow\infty$ and using that $t_0$ is a point of differentiability by assumption, we obtain that $$M'(t_0) \leq \partial_t\theta(t_0,x_0) = -u(t_0,x_0) \cdot \nabla\theta(t_0,x_0) - \kappa(-\Delta)^{\gamma/2}\theta(t_0,x_0).$$ The first term on the RHS of the preceding equality is nonpositive since $\theta(t_0)$ is maximal at $x_0$. The first term on the RHS is also nonpositive for the same reason since $$(-\Delta)^{\gamma/2}\theta(t_0,x_0) = C_\gamma PV\int_{\mathbb{R}^2}\frac{\theta(t_0,x_0)-\theta(t_0,y)}{|x-y|^{2+\gamma}}dy.$$ Hence, $M'(t_0)\leq 0$. Note that by considering a sequence $h_j$ of negative numbers, one can show that $M'(t_0) = -\kappa(-\Delta)^{\gamma/2}\theta(t_0,x_0)$. That $m$ is nondecreasing follows by replacing $\theta$ by $-\theta$ in the preceding argument.