Rademacher's theorem

The set of Rademacher , named after the German mathematician Hans Rademacher , is a set of Analysis on Lipschitz functions.

statement

Let be natural numbers , an open subset of a Euclidean space and finally a Lipschitz continuous function . Then it is (totally) differentiable almost everywhere . ${\ displaystyle n, m \ in \ mathbb {N}}$ ${\ displaystyle U \ subseteq \ mathbb {R} ^ {n}}$ ${\ displaystyle f \ colon U \ to \ mathbb {R} ^ {m}}$ ${\ displaystyle f}$

That is, the set of all points in which is not differentiable is a Lebesgue null set . ${\ displaystyle f}$

generalization

There is a generalization for functions , where now denotes any metric space . ${\ displaystyle f \ colon U \ to (X; d_ {X})}$ ${\ displaystyle X}$

At first, however, it is not clear how the above sentence can be transferred to this case, because a metric space does not a priori also have a linear structure .

If one understands as a function between normalized spaces and takes the Fréchet differentiability as a basis, then the proposition is even wrong: ${\ displaystyle f}$

The classic counter-example here is the function . Where denotes the characteristic function of the sub-interval .

{\ displaystyle \ Phi \ colon [0; 1] \ to L ^ {1} ([0; 1]), \ t \ mapsto \ chi _ {[0; t]}}

{\ displaystyle \ chi _ {[0; t]}}

{\ displaystyle [0; t]}

It applies to any :

{\ displaystyle \ 1 \ geq y \ geq x \ geq 0}

{\ displaystyle \ | \ Phi (y) - \ Phi (x) \ | _ {L ^ {1}} = \ int _ {0} ^ {1} | \ chi _ {[0, y]} (t ) - \ chi _ {[0, x]} (t) | \, dt = \ int _ {x} ^ {y} 1 \ dt = | yx | \ ,.}

The L ¹ norm denotes . That is to say, it is an isometry and thus even more Lipschitz-continuous, but it can be shown that Fréchet is nowhere differentiable.

{\ displaystyle \ |. \ | _ {L ^ {1}}}

{\ displaystyle \ Phi}

{\ displaystyle \ Phi}

The German mathematician Bernd Kirchheim has now been able to generalize Rademacher's theorem in a different sense:

If a function is Lipschitz continuous from a Euclidean to a metric space, it is metrically differentiable almost everywhere .

Individual evidence

↑ Juha Heinonen , Lectures on Lipschitz Analysis (PDF; 481 kB) , Lectures at the 14th Jyväskylä Summer School in August 2004. (Theorem by Rademacher including a proof: pp. 18ff.) Retrieved on June 12, 2012.
↑ Bernd Kirchheim: Rectifiable metric spaces: Local structure and regularity of the Hausdorff measure ; Quoted from: Proceedings of the American Mathematical Society : Volume 121, Number 1, May 1994. Retrieved June 12, 2012.

[1] Juha Heinonen , Lectures on Lipschitz Analysis (PDF; 481 kB) , Lectures at the 14th Jyväskylä Summer School in August 2004. (Theorem by Rademacher including a proof: pp. 18ff.) Retrieved on June 12, 2012.

[2] Bernd Kirchheim: Rectifiable metric spaces: Local structure and regularity of the Hausdorff measure ; Quoted from: Proceedings of the American Mathematical Society : Volume 121, Number 1, May 1994. Retrieved June 12, 2012.