(Extension theorem) Let
are bounded and
.
Assume that
admits a locally continuously differentiable parametrization. Then there
exists a bounded linear "extension"
operator
such that for any
a.s. in
and the support of
lies within
.

Proof

The
may be
-expanded
across a flat boundary
by the reflection of the
form
If the boundary is not flat then there exists a change of variables that makes
it locally flat. Then such procedure extends globally using the partition of
unity (see the proof of the proposition
(
Global approximation by
smooth functions
) for an example of the technique). The partition of unity
insures that the support of the result is localized.