Proof
We seek such family using the standard mollifiers, see the definition
(
Standard mollifier definition
).
We introduce
:
we substitute
,
:
Note that the argument
lies outside of
for
close to the boundary
.
We solve this problem via the proposition
(
Extension theorem
): we extend the
to a slightly larger set
and perform the convolution using the extended functions.

We initially assume that the functions
are smooth and
estimate
note that
,
Hence,
Since the support of the extended function
lies strictly within
,
we can drop the small shift
in the integral
without changing the
integral:
Hence,
According to the proposition
(
Approximation by smooth
functions
) this estimate extends to the functions
.

Finally, we use the boundedness of
(and
)
and the formula (
Holder inequality
) with
:
Hence,
Thus we established
that

Next we intend to the other values of
using the formula (
Lp
interpolation
):

We aim to establish that
dominates
.
Hence, we match
,
and
.
Then
Hence,
The sequences
are bounded in
.
Hence, by the proposition
(
Gagliardo-Nirenberg-Sobolev
inequality
) these are bounded in
.
Therefore, the
term in the last estimate is bounded and,
consequently,

To see the second part of the statement we verify that
and
are bounded for every
: