In the previous entry, we showed the following
Theorem. Let and be two smooth functions on satisfying
Suppose that is bounded and also and
As suggested in an earlier entry, in this topic, we show that the following limit
for some good .
It is worth recalling that if we denote by the following quantity
This is about to say that it suffice to prove
exists. Amazingly, the following
Theorem. If satisfies as then
exists and is finite.
It suffices to show that
exists and is finite. Observe that
We show that
For each we consider
Then we decompose into where are integrals on the region , respectively.
- The case of .
We start with , then
- The case of . For large , we observe that
We can further estimate. We get
as . Therefore, as .
- The case of . In this situation, we first fix . Then
as . Then we take .
We are going to prove another result
for some positive constant .