Firstly, we prove the following representation formula on
, which is of its independent interest.
Lemma 3.1.
Let
and
. Then,
where
is a diagonal matrix given by
Proof.
We argue as in the proof of Theorem 1.2 in [22]. Since
,
Therefore,
Notice that
we have, by (3.4),
To finish the proof, it is enough to show that
vanishes. Notice that the operator
annihilates functions of
, and, for
, the integrand above is absolutely integrable. We have, for any
, though integration by parts,
Let
. By dominated convergence theorem,
The proof is therefore completed.
We now prove the following Hardy inequalities on
.
Theorem 3.2.
Let
and
. There holds, for all
,
Proof.
Set
with
. Replacing
by
in Lemma 3.1, we obtain, for any
,
It is easy to check that the following equations hold
Therefore, by (3.11),
By dominated convergence, letting
, we have
By Hölder's inequality,
Canceling and raising both sides to the power
, we obtain
Set
. Then,
, and we get (3.11).
Remark 3.3.
Notice that
, we have, by Theorem 3.2, for all
,
From inequality (3.17), we have the following corollary which generalizes the result of [21] when
and
.
Corollary 3.4.
Let
,
and
. There holds, for all
,
Proof.
Since
,
We have, by inequality (3.17),