Bounded mean oscillation

The BMO space is an object from the harmonic analysis, a branch of mathematics. The acronym BMO stands for "bounded mean oscillation ". The function space BMO was introduced in 1961 by Fritz John and Louis Nirenberg. This space is a dual space to the real Hardy space (Charles Fefferman, Elias Stein 1972).

Definitions

Sharp function

Let be a locally integrable function, defined by

Where the supremum is taken over all balls containing. With the averaging integral is

Referred to.

BMO space

A locally integrable function is called BMO function if is limited. To obtain a standard function in this space, it identifies all constant functions with each other and sets

If you were the constant functions do not identify with one another, so would only be a semi-norm, that is not definite. This standard is BMO - space to a Banach space. Examples of BMO functions are all limited measurable functions and a polynomial P, which is not identical to zero.

Duality of H1 and BMO

Charles Fefferman showed in 1971 that the BMO space is a dual space of, the real Hardy space with p = 1,. The pairing between and is given by

Then the picture is a Banach space isomorphism ( not isometric ), in this sense, is the dual space of.

However, the above integral expression must be carefully defined, since this integral generally does not converge absolutely. However, there is a dense subspace on which the integral converges absolutely. With the help of the theorem of Hahn- Banach we can resume functional on all. As a room, you can choose the space of H1 - functions with compact support and with. This is exactly the subspace which has a finite atomic decomposition. An important consequence which follows from the proof of duality is the following inequality, which is valid for and:

Here, the non- tangential maximum function.