We now prove the following result
Theorem. Let and be two smooth functions on satisfying
Suppose that is bounded and also and
It is easy to see that
We claim that
To see this, we need only to verify that
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
- For the case of , we fix and observe that
as . Therefore
- For the case of , we firstly fix and consider sufficiently large such that
This is possible since
We now let and then to get the desired result.
From above, we deduce that
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].