Firstly, we prove the following representation formula on , which is of its independent interest.
Let and . Then,
where is a diagonal matrix given by
We argue as in the proof of Theorem 1.2 in . Since ,
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 .
Let and . There holds, for all ,
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).
Notice that , we have, by Theorem 3.2, for all ,
From inequality (3.17), we have the following corollary which generalizes the result of  when and .
Let , and . There holds, for all ,
We have, by inequality (3.17),