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?



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