# Monthly Archives: August 2016

## An easy proof of the John-Nirenberg Inequality

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 