We now prove the following result

Theorem. Let and be two smooth functions on satisfying.

Suppose that is bounded and also and

.

Then

.

*Proof*.

Let

.

It is easy to see that

.

We claim that

.

To see this, we need only to verify that

as .

Decompose into three domains

and write where are integrals over these domains, respectively. We consider , then we estimate as follows.

- For the case of , note that
which implies

as .

- For the case of , we fix and observe that
which implies

as . Therefore

as .

- For the case of , we firstly fix and consider sufficiently large such that
.

This is possible since

and

provided

since

.

Thus

We now let and then to get the desired result.

From above, we deduce that

as .

Now by using the potential analysis together with asymtotic behavior of we deduce that

which completes the proof.

For other estimates, we refer the reader to a paper due to Wang and Zhu published in Duke Math. J. (2000) [here].

## Leave a Reply