Bloch group

2013-01-16T18:56:07Z

129.2.56.35: /* Relations between K3 and Bloch group */

{{Refimprove|date=September 2011}}
In [[mathematical analysis]], a '''metric differential''' is a generalization of a [[derivative]] for a [[Lipschitz_continuity|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|Rademarcher's theorem]] to metric space-valued Lipschitz functions.

==Discussion==

[[Rademacher's theorem]] states that a Lipschitz map ''f'' : '''R'''''n'' → '''R'''''m'' is differentiable [[almost everywhere]] in '''R'''''n''; 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'''''n'' 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'' : [0,1] → ''L''1([0,1]), mapping the unit interval into the [[Lp_space|space of integrable functions]], defined by ''f''(''x'') = ''χ''[0,''x''], this function is Lipschitz (and in fact, an [[isometry]]) since, if 0 ≤ ''x'' ≤ ''y''≤ 1, then

:<math>|f(x)-f(y)|=\int_0^1 |\chi_{[0,x]}(t)-\chi_{[0,y]}(t)|\,dt = \int_x^y \, dt = |x-y|,</math>

but one can verify that lim''h''→0(''f''(''x'' + ''h'') −  ''f''(''x''))/''h'' does not converge to an ''L''1 function for any ''x'' in [0,1], 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'''n → ''X'' is the metric differential of ''f'' at a point ''z'' in '''R'''''n'' which is a function on '''R'''''n'' defined by the limit

: <math> MD(f,z)(x)=\lim_{r\rightarrow 0} \frac{d_{X}(f(z+rx),f(z))}{r} </math>
whenever the limit exists (here ''d'' ''X'' denotes the metric on ''X'').

A theorem due to Bernd Kirchheim<ref>{{cite journal
| last = Kirchheim | first = Bernd | authorlink = Bernd Kirchheim
| title = Rectifiable metric spaces: local structure and regularity of the Hausdorff measure
| journal =Proc. of the Am. Math. Soc.| volume = 121 | pages = 113–124 | year = 1994
}}</ref> states that a Rademacher theorem in terms of metric differentials holds: for almost every ''z'' in '''R'''''n'', MD(''f'', ''z'') is a [[seminorm]] and

: <math> d_X(f(x),f(y)) - MD(f,z)(x-y) = o(|x-z|+|y-z|). \, </math>

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

== References ==
{{Reflist}}

[[Category:Lipschitz maps]]
[[Category:Mathematical analysis]]

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Bloch group