# Extension of Lipschitz functions that preserve the Frobenius norm of the Jacobian

Let $$n,m\ge 1$$ be integers and let $$f:E\to R^m$$ be $$L$$-Lipschitz for some subset $$E\subset R^n$$.

Kirszbraun's theorem, https://en.wikipedia.org/wiki/Kirszbraun_theorem, states that there exists another $$L$$-Lipschitz function $$\bar f:R^n\to R^m$$ such that $$f=\bar f$$ on $$E$$ and $$\bar f$$ is called an extension of $$f$$.

I am wondering about similar extensions of $$f$$ that maintain the Frobenius norm of the gradient (instead of the operator norm in Kirszbraun's theorem).

1. Assume that $$E$$ is open. By Rademacher's theorem, the Frechet derivative $$D(x)$$ (viewed as the matrix of size $$m\times n$$ such that $$f(x+h)=f(x) + D(x)h + o(\|h\|)$$) exists at almost every $$x$$.
Prove or disprove that there always exists an extension $$\bar f$$ such that the essential supremum of the Frobenius norm $$\|\bar D(x)\|_F$$ of the the Frechet derivative $$\bar D(x)$$ of $$\bar f$$ is no more than the essential supremum of $$\|D(x)\|_F$$.

Edit: the above question is actually not well posed: if for instance $$E_1,...,E_k$$ are disjoint open sets and $$f$$ is constant on each open set then $$D(x)=0$$ on $$E=E_1\cup...\cup E_k$$. This is because we lack a global'' property akin to bounds on the Frobenius norm of $$D(x)$$ (a property similar $$\|f(x)-f(y)\|\le L \|x-y\|$$ which encodes $$\|D(x)\|_{op}\le L$$ in a global fashion). This leads to these possibly more meaningful questions:

1. How to encode the local property $$\|D(x)\|_{F}\le K$$ with a global property, so that $$\|D(x)\|_{F}\le K$$ almost everywhere for $$f:R^n\to R^m$$ and if and only if the global property holds? Prove or disprove that such global property exists.

2. (provided that such global property exists) If $$f$$ satisfies the global property on a subset $$E\subset R^n$$, is there an extension $$\bar f: R^n\to R^m$$ of $$f$$ that satisfies the global property?