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Metric differential

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In mathematical analysis, a metric differential is a generalization of a derivative for a Lipschitz continuous function defined on a Euclidean space and taking values in an arbitrary metric space. With this definition of a derivative, one can generalize Rademacher's theorem to metric space-valued Lipschitz functions.

Discussion

Rademacher's theorem states that a Lipschitz map f : R → R is differentiable almost everywhere in R; in other words, for almost every x, f is approximately linear in any sufficiently small range of x. If f is a function from a Euclidean space R that takes values instead in a metric space X, it doesn't immediately make sense to talk about differentiability since X has no linear structure a priori. Even if you assume that X is a Banach space and ask whether a Fréchet derivative exists almost everywhere, this does not hold. For example, consider the function f :  → L(), mapping the unit interval into the space of integrable functions, defined by f(x) = χ, this function is Lipschitz (and in fact, an isometry) since, if 0 ≤ x ≤ y≤ 1, then

| f ( x ) − f ( y ) | = ∫ 0 1 | χ [ 0 , x ] ( t ) − χ [ 0 , y ] ( t ) | d t = ∫ x y d t = | x − y | , {\displaystyle |f(x)-f(y)|=\int _{0}^{1}|\chi _{}(t)-\chi _{}(t)|\,dt=\int _{x}^{y}\,dt=|x-y|,}

but one can verify that limh→0(f(x + h) −  f(x))/h does not converge to an L function for any x in , so it is not differentiable anywhere.

However, if you look at Rademacher's theorem as a statement about how a Lipschitz function stabilizes as you zoom in on almost every point, then such a theorem exists but is stated in terms of the metric properties of f instead of its linear properties.

Definition and existence of the metric differential

A substitute for a derivative of f:R → X is the metric differential of f at a point z in R which is a function on R defined by the limit

M D ( f , z ) ( x ) = lim r → 0 d X ( f ( z + r x ) , f ( z ) ) r {\displaystyle MD(f,z)(x)=\lim _{r\rightarrow 0}{\frac {d_{X}(f(z+rx),f(z))}{r}}}

whenever the limit exists (here d X denotes the metric on X).

A theorem due to Bernd Kirchheim states that a Rademacher theorem in terms of metric differentials holds: for almost every z in R, MD(f, z) is a seminorm and

d X ( f ( x ) , f ( y ) ) − M D ( f , z ) ( x − y ) = o ( | x − z | + | y − z | ) . {\displaystyle d_{X}(f(x),f(y))-MD(f,z)(x-y)=o(|x-z|+|y-z|).}

The little-o notation employed here means that, at values very close to z, the function f is approximately an isometry from R with respect to the seminorm MD(f, z) into the metric space X.

References

  1. Kirchheim, Bernd (1994). "Rectifiable metric spaces: local structure and regularity of the Hausdorff measure". Proceedings of the American Mathematical Society. 121: 113–124. doi:10.1090/S0002-9939-1994-1189747-7.
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