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.


Blog at