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(eg Sobolev embedding, Lebesgue differentiation theorem.)

Answer 1

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 2

• There is no path isometry $\mathbb R^2\to\mathbb R$;
• There is no path isometry $(\mathbb R^2,\ell_p)\to\mathbb R^n$ for $p\not=2$.

Answer 3

Sir can you post the proof of there is no path isometry from R2 to R and the next statement ?

Answer 4

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 http://arxiv.org/abs/1212.0247 .)

A few years ago with Hans Lindblad in http://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 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.