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.
According to Nirenberg (1985, p. 703 and p. 707),[1] 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
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),[2] 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[3] of the duality between BMO and the Hardy space
, 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.
The mean oscillation of a locally integrable function
whose mean oscillation supremum, taken over the set of all cubes
The supremum of the mean oscillation is called the BMO norm of
as the integration domains on which the mean oscillation is calculated, is not mandatory: Wiegerinck (2001) uses balls instead and, as remarked by Stein (1993, p. 140), in doing so a perfectly equivalent definition of functions of bounded mean oscillation arises.
In fact, using the John-Nirenberg Inequality, we can prove that Constant functions have zero mean oscillation, therefore functions differing for a constant
does not oscillate very much when computing it over cubes close to each other in position and scale.
are dyadic cubes such that their boundaries touch and the side length of
is given by though some care is needed in defining this integral, as it does not in general converge absolutely.
The John–Nirenberg Inequality is an estimate that governs how far a function of bounded mean oscillation may deviate from its average by a certain amount.
Conversely, if this inequality holds over all cubes with some constant
the following two-sided inequality holds When the dimension of the ambient space is 1, the space BMO can be seen as a linear subspace of harmonic functions on the unit disk and plays a major role in the theory of Hardy spaces: by using definition 2, it is possible to define the BMO(T) space on the unit circle as the space of functions f : T → R such that i.e. such that its mean oscillation over every arc I of the unit circle[10] is bounded.
An Analytic function on the unit disk is said to belong to the Harmonic BMO or in the BMOH space if and only if it is the Poisson integral of a BMO(T) function.
Charles Fefferman in his original work proved that the real BMO space is dual to the real valued harmonic Hardy space on the upper half-space Rn × (0, ∞].
[11] In the theory of Complex and Harmonic analysis on the unit disk, his result is stated as follows.
[12] Let Hp(D) be the Analytic Hardy space on the unit Disc.
For p = 1 we identify (H1)* with BMOA by pairing f ∈ H1(D) and g ∈ BMOA using the anti-linear transformation Tg Notice that although the limit always exists for an H1 function f and Tg is an element of the dual space (H1)*, since the transformation is anti-linear, we don't have an isometric isomorphism between (H1)* and BMOA.
However one can obtain an isometry if they consider a kind of space of conjugate BMOA functions.
The space VMO of functions of vanishing mean oscillation is the closure in BMO of the continuous functions that vanish at infinity.
It can also be defined as the space of functions whose "mean oscillations" on cubes Q are not only bounded, but also tend to zero uniformly as the radius of the cube Q tends to 0 or ∞.
[13] A locally integrable function f on R is BMO if and only if it can be written as where fi ∈ L∞, α is a constant and H is the Hilbert transform.
Similarly f is VMO if and only if it can be represented in the above form with fi bounded uniformly continuous functions on R.[14] Let Δ denote the set of dyadic cubes in Rn.
The space dyadic BMO, written BMOd is the space of functions satisfying the same inequality as for BMO functions, only that the supremum is over all dyadic cubes.
However, if a function f is such that ||f(•−x)||BMOd ≤ C for all x in Rn for some C > 0, then by the one-third trick f is also in BMO.
By duality, H1(Tn ) is the sum of n+1 translation of dyadic H1.