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),