(Extension theorem) Let
are bounded and
admits a locally continuously differentiable parametrization. Then there
exists a bounded linear "extension"
such that for any
and the support of
across a flat boundary
by the reflection of the
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
) for an example of the technique). The partition of unity
insures that the support of the result is localized.