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

(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