# An easy proof of the John-Nirenberg Inequality

By | August 31, 2016

One of fundamental theorem in the space of bounded mean oscilation is the John-Nirenberg inequality.

Theorem(John-Nirenberg). Let . Then

for all where , .

In particular,

where .

There are several proofs. Stein(1994 Monograph) gives a proof by using -BMO duality inequality. Here we give another proof based on dyadic decomposition given by Martell in the Lecture of MSRI.

Proof. We will use dyadic decomposition idea. By scaling, without loss of generality, we may assume . Define

We want to show that . First take . Then

for any . Subdivide dyadically and stop when . Collect these cubes , where we have stopped (it could be ). Note that , where denotes the family of all dyadic cubes of .

Now we introduce the following type of dyadic maximal function

where . Then by definition of , we have

For almost every we have

So

(1)

Let be a parent cube of Then by definition of cube

(2)

Now

For , , we have

Hence .

Since , by (1) we have

Then by (2), we have

Hence for , we obtain

By taking supremum over , we have

Put . Note that for all . So for , we see that

Note that

So for , we obtain for , Since we have

we see that for Hence, for , we have

Iterate this procedure. So we obtain the desired claim. This leads to

This proves the first part. For the second part, use