Abstract
Let
C^{m, \omega} ( \mathbb{R}^n)
be the space of functions on
\mathbb{R}^n
whose
m^{\sf th}
derivatives have modulus of continuity
\omega
. For
E \subset \mathbb{R}^n
, let
C^{m , \omega} (E)
be the space of all restrictions to
E
of functions in
C^{m , \omega} ( \mathbb{R}^n)
. We show that there exists a bounded linear operator
T: C^{m , \omega} ( E ) \rightarrow C^{m , \omega } ( \mathbb{R}^n)
such that, for any
f \in C^{m , \omega} ( E )
, we have
T f = f
on
E
.