One of fundamental theorem in the space of bounded mean oscilation is the John-Nirenberg inequality.
Theorem(John, Nirenberg). Let . Then
for all 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
For , , we have
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
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