# Ngô Quốc Anh

## February 25, 2015

### Continuous functions on subsets can be extended to the whole space: The Kirzbraun-Pucci theorem

Filed under: Uncategorized — Ngô Quốc Anh @ 1:22

Let $f$ be a continuous function defined on a set $E \subset \mathbb R^N$ with values in $\mathbb R$ and with modulus of continuity

$\displaystyle \omega_f (s) := \sup_{|x-y|\leqslant s,x,y\in E} |f(x) - f(y)| \quad s>0.$

Obviously, the function $s \mapsto \omega_f(s)$ is nonnegative and nondecreasing in $[0,+\infty)$.

Our first assumption is that $\omega_f$ is bounded from above in $[0, \infty)$ by some increasing, affine function; that is to say there exists some $a,b \in \mathbb R^+$ such that

$\displaystyle \omega_f (s) \leqslant a s +b \quad \forall s \geqslant 0$.

Associated with $\omega_f$ having the above first assumption is the concave modulus of continuity of $f$, i.e. some smallest concave function $c_f$ lies above $\omega_f$. Such the function $c_f$ can be easily constructed using the following

$\displaystyle c_f (s) = \inf_\ell \{\ell(s) : \ell \text{ is affine and } \ell \geqslant \omega_f \text{ in } [0,+\infty)\}.$

As can be easily seen, once $\omega_f$ can be bounded from above by some affine function, the concave modulus of continuity of $f$ exists and is well-defined.

By definition and the monotonicity of $\omega_f$, we obtain

$\displaystyle |f(x)-f(y)| \leqslant \omega_f (|x-y|) \leqslant c_f (|x-y|).$

In this note, we prove the following extension theorem.

Theorem (Kirzbraun-Pucci). Let $f$ be a real-valued, uniformly continuous function on a set $E \subset \mathbb R^N$ with modulus of continuity $\omega_f$ satisfying the first assumption. There exists a continuous function $\widetilde f$ defined on $\mathbb R^N$ that coincides with $f$ on $E$. Moreover, $f$ and $\widetilde f$ have the same concave modulus of continuity $c_f$ and

$\displaystyle \sup_{\mathbb R^N} \widetilde f = \sup_E f, \quad \inf_{\mathbb R^N} \widetilde f = \inf_E f.$