(Riesz potential bound 1) Let
and either
and
or
and
then

Proof

First, we consider the case
and
.
According to the formula (
Holder inequality
)
We make the change to polar coordinates in the first
integral:
for some function
and
estimate
We need the expression
to be greater then
for the last integral to exist. We
require
and transform the above into a condition for
:
We continue estimation
of
thus
In case
and
we take more direct
route:

Proposition

(Remainder bound 1) Let
and either
and
or
and
then

(Sobolev inequality 2) Suppose
is star shaped with respect to the ball
.
Let
and either
,
or
,
then

Proof

We apply the proposition (
Sobolev
inequality
) in the following form
for all
s.t.
.
Thus we need either
or
.
We set
and arrive to either
,
or
,
as required in the proposition. Therefore, we arrive
to

Proposition

(Riesz potential bound 2) Let
,
,
and
then

Proof

For
we
estimate
We apply the formula (
Holder inequality 3
)
with
and
to the internal
integral.
We estimate
,
.
We change the order of integration (see the proposition
(
Fubini
theorem
)).
Note that
.

Thus

For
we
estimate
We change the order of
integration.
For
we
estimate