Bounded mean oscillation

In harmonic analysis in mathematics, a function of bounded mean oscillation, also known as a BMO function, is a real-valued function whose mean oscillation is bounded (finite). The space of functions of bounded mean oscillation (BMO) is a function space that, in some precise sense, plays the same role in the theory of Hardy spaces Hp that the space L∞ of essentially bounded functions plays in the theory of Lp-spaces. It is also called John–Nirenberg space, after Fritz John and Louis Nirenberg who introduced and studied it for the first time.

Historical note

According to Nirenberg (1985, p. 703 and p. 707), the space of functions of bounded mean oscillation was introduced by John (1961, pp. 410–411) in connection with his studies of mappings from a bounded set

Ω

{\displaystyle \Omega }

belonging to

R

n

{\displaystyle \mathbb {R} ^{n}}

into

R

n

{\displaystyle \mathbb {R} ^{n}}

and the corresponding problems arising from elasticity theory, precisely from the concept of elastic strain: the basic notation was introduced in a closely following paper by John & Nirenberg (1961), where several properties of this function spaces were proved. The next important step in the development of the theory was the proof by Charles Fefferman of the duality between BMO and the Hardy space

H

1

{\displaystyle H^{1}}

, in the noted paper Fefferman & Stein 1972: a constructive proof of this result, introducing new methods and starting a further development of the theory, was given by Akihito Uchiyama.

Definition

Definition 1. The mean oscillation of a locally integrable function

u

{\displaystyle u}

over a hypercube

Q

{\displaystyle Q}

in

R

n

{\displaystyle \mathbb {R} ^{n}}

is defined as the value of the following integral:

1

|

Q

|

∫

Q

|

u

(

y

)

−

u

Q

|

d

y

{\displaystyle {\frac {1}{|Q|}}\int _{Q}|u(y)-u_{Q}|\,\mathrm {d} y}

where

|

Q

|

{\displaystyle |Q|}

is the volume of

Q

{\displaystyle Q}

, i.e.