이 페이지는 제가 Interpolation theorem들을 하나씩 배울 때마다 업로드할 예정입니다. (This page will be updated frequently when I learned some interpolation theorems.)
조금 rough하게 말하면, interpolation theorem이란 가 이고 일 때 그 사이 공간 ()에 들어갈 조건이나, 의 -norm의 크기를 재는 정리들을 말합니다.
1. Basic Interpolation Theorems
Theorem. If for some , then for all and that , where is such that .
Proof. We first assume . Assume for some . So .
Set and .
Then by applying Hölder’s inequality to , we have
So by taking to both sides, we get
The equality condition holds when for some nonnegative a.e.
For , note that we have . So
Since , we have
Theorem. Let and let , where is -finite measure space. Then for all and
with the interpretation that .
Proof. First, assume . Then by definition of weak -norm, we have
We estimate as follow:
So taking and calculate some minor things, we get
3. The Complex Interpolation Method: The Riesz-Thorine Interpolation Theorem
Theorem (The Riesz-Thorine Interpolation Theorem) Let and be two -finite measure spaces. Let be a linear operator defined on the set of all finitely simple functions on and taking values in the set of measurable functions on . Let and assume that
for all finitely simple functions on . Then for all , we have
for all finitely simple functions on , where
Consequently, when , by density, has a unique bounded extension from to .
Proof: We divide our proof into three steps.
Step 1. Observe that for , it is clear by Theorem
So we may assume and we further assume that for .
Let denote the space of all finite simple function on . Then clearly, and . So is . We will show that for any , we have
By considering the dual space of , we have
where is a conjugate exponent of .
We further assume that with . We will show the following claim in Step 2:
- If and , then for all such that .
Step 2. Let and with and . Let and be the canonical representation. Write , . Also, let
thus and for .
Fix . Since , . So we may define
If , we define
while if , we define for all .
We first show our claim when . Finally, we set
so that is an entire holomorphic function of that is bounded in the strip . Since , by Three lines lemma, it suffices to show for and for to show the claim.
First, note that
for . This shows
Therefore, by Hölder’s inequality, we have
Similarly, we get
The case is similar.
So we show the claim.
Step 3. From Step 2, we have shown that for . Then has a unique extension to satisfying the same estimate there. It remains to show that this extension is itself.
Given , choose a sequence in such that and pointwise. Also, let , , , , and . Then if , we have , and by DCT, , , and . Hence, and . So by passing to a suitable subsequence, we may assume that a.e. and a.e. But then a.e. So by Fatou’s lemma, we get
and we are done.
4. The Real Interpolation Method: The Marcinkiewicz Interpolation Theorem
Definition 1 Let and be two measure spaces. Suppose we are given an operator . Operators that map to are called of strong type and operators that map to are called weak type .
Definition Let be an operator defined on a vector space of complex-valued measurable functions on a measure space and taking values in the set of all complex-valued finite almost everywhere measurable functions on a measure space . Then is called linear if for all in the domain of and all we have
is called sublinear if for all in the domain of and all we have
is called quasi-linear if for all in the domain of and all we have
for some constant .
We are now ready to state the Marcinkiewicz Interpolation Theorem.
Theorem (Marcinkiewicz Interpolation Theorem) Let be a -finite measure space, let be another measure space, and let . Let be a sublinear operator defined on and taking values in the space of measurable functions on . Assume that there exist such that
Then for all and for all , we have the estimate
Proof: We divide our proof into two cases: when and .
Let and let . Split to be , by defining
The constant will be determined later.
NNote that and we estimate ,
Since is sublinear by subadditivity we get
So from this,
So by definition of weak norm, we get
Similarly, we get
Finally, using Fubini’s theorem since is -finite,
Choose so that
Then we get
Finally, using Fubini’s theorem since is -finite, we have
Choose so that
Then we get
Write , where
where provided we choose . So has measure zero. Therefore,
Since maps to with norm at most , it follows that
Therefore, we obtain
This proves the theorem with constant