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

Set

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

Thus,

where

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 1Let and be two measure spaces. Suppose we are given an operator . Operators that map to are called ofstrong typeand operators that map to are calledweak type.

DefinitionLet 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 calledlinearif for all in the domain of and all we haveis called

sublinearif for all in the domain of and all we haveis called

quasi-linearif 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

where

*Proof:* We divide our proof into two cases: when and .

*Case 1. *

Let and let . Split to be , by defining

and

The constant will be determined later.

NNote that and we estimate ,

as follows:

Similarly,

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,

we have

Choose so that

Then we get

where

Finally, using Fubini’s theorem since is -finite, we have

Choose so that

Then we get

where

*Case 2*.

Write , where

We have

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