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 *H ^{p}* that the space

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 Ω belonging to **R**^{n} into **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),^{[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 *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.^{[4]}

Definition 1. The **mean oscillation** of a locally integrable function *u* over a hypercube^{[5]} *Q* in **R**^{n} is defined as the value of the following integral:

where

*Q*is the volume of*Q*, i.e. its Lebesgue measure*u*is the average value of_{Q}*u*on the cube*Q*, i.e.

Definition 2. A **BMO function** is a locally integrable function *u* whose mean oscillation supremum, taken over the set of all cubes *Q* contained in **R**^{n}, is finite.

**Note 1**. The supremum of the mean oscillation is called the **BMO norm** of *u*.^{[6]} and is denoted by *u*_{BMO} (and in some instances it is also denoted *u*_{∗}).

**Note 2**. The use of cubes *Q* in **R**^{n} 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.

- The universally adopted notation used for the set of BMO functions on a given domain Ω is
**BMO**(Ω): when Ω =**R**^{n},**BMO**(**R**^{n}) simply symbolized as**BMO**. - The
**BMO norm**of a given BMO function*u*is denoted by*u*_{BMO}: in some instances, it is also denoted as*u*_{∗}.

BMO functions are locally *L ^{p}* if 0 <

- .

Constant functions have zero mean oscillation, therefore functions differing for a constant *c* > 0 can share the same BMO norm value even if their difference is not zero almost everywhere. Therefore, the function *u*_{BMO} is properly a norm on quotient space of BMO functions modulo the space of constant functions on the domain considered.

As the name suggests, the mean or average of a function in BMO does not oscillate very much when computing it over cubes close to each other in position and scale. Precisely, if *Q* and *R* are dyadic cubes such that their boundaries touch and the side length of *Q* is no less than one-half the side length of *R* (and vice versa), then

where *C* > 0 is some universal constant. This property is, in fact, equivalent to *f* being in BMO, that is, if *f* is a locally integrable function such that *f _{R}*−

Fefferman (1971) showed that the BMO space is dual to *H*^{1}, the Hardy space with *p* = 1.^{[8]} The pairing between *f* ∈ *H*^{1} and *g* ∈ BMO 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.

For eachf∈ BMO(R^{n}) there are constantsc_{1},c_{2}> 0 , such that for any cubeQinR^{n},

Conversely, if this inequality holds over all cubes with some constant *C* in place of *f*_{BMO}, then *f* is in BMO with norm at most a constant times *C*.

The John–Nirenberg inequality can actually give more information than just the BMO norm of a function. For a locally integrable function *f*, let *A*(*f*) be the infimal *A*>0 for which

The John–Nirenberg inequality implies that *A*(*f*) ≤ C*f*_{BMO} for some universal constant *C*. For an *L*^{∞} function, however, the above inequality will hold for all *A* > 0. In other words, *A*(*f*) = 0 if *f* is in L^{∞}. Hence the constant *A*(*f*) gives us a way of measuring how far a function in BMO is from the subspace *L*^{∞}. This statement can be made more precise:^{[9]} there is a constant *C*, depending only on the dimension *n*, such that for any function *f* ∈ BMO(**R**^{n}) 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. Here as before *f _{I}* is the mean value of f over the arc I.

Definition 3. 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. Therefore, BMOH is the space of all functions *u* with the form:

equipped with the norm:

The subspace of analytic functions belonging BMOH is called the **Analytic BMO space** or the **BMOA space**.

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 **R**^{n} × (0, ∞].^{[11]} In the theory of Complex and Harmonic analysis on the unit disk, his result is stated as follows.^{[12]} Let *H ^{p}*(

Notice that although the limit always exists for an *H*^{1} function f and *T _{g}* is an element of the dual space (

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 ∞. The space VMO is a sort of Hardy space analogue of the space of continuous functions vanishing at infinity, and in particular the real valued harmonic Hardy space *H*^{1} is the dual of VMO.^{[13]}

A locally integrable function *f* on **R** is BMO if and only if it can be written as

where *f _{i}* ∈

The BMO norm is then equivalent to the infimum of over all such representations.

Similarly *f* is VMO if and only if it can be represented in the above form with *f _{i}* bounded uniformly continuous functions on

Let *Δ* denote the set of dyadic cubes in **R**^{n}. The space **dyadic BMO**, written BMO_{d} is the space of functions satisfying the same inequality as for BMO functions, only that the supremum is over all dyadic cubes. This supremum is sometimes denoted *• _{BMOd}*.

This space properly contains BMO. In particular, the function *log(x)χ _{[0,∞)}* is a function that is in dyadic BMO but not in BMO. However, if a function

Although dyadic BMO is a much narrower class than BMO, many theorems that are true for BMO are much simpler to prove for dyadic BMO, and in some cases one can recover the original BMO theorems by proving them first in the special dyadic case.^{[15]}

Examples of BMO functions include the following:

- All bounded (measurable) functions. If
*f*is in L^{∞}, then*f*_{BMO}≤ 2f_{∞}:^{[16]}however, the converse is not true as the following example shows. - The function log(
*P*) for any polynomial*P*that is not identically zero: in particular, this is true also for*P*(*x*) =*x*.^{[16]} - If
*w*is an*A*_{∞}weight, then log(*w*) is BMO. Conversely, if*f*is BMO, then*e*^{δf}is an*A*_{∞}weight for δ>0 small enough: this fact is a consequence of the John–Nirenberg Inequality.^{[17]}

**^**Aside with the collected papers of Fritz John, a general reference for the theory of functions of bounded mean oscillation, with also many (short) historical notes, is the noted book by Stein (1993, chapter IV).**^**The paper (John 1961) just precedes the paper (John & Nirenberg 1961) in volume 14 of the Communications on Pure and Applied Mathematics.**^**Elias Stein credits only Fefferman for the discovery of this fact: see (Stein 1993, p. 139).**^**See his proof in the paper Uchiyama 1982.**^**When*n*= 3 or*n*= 2,*Q*is respectively a cube or a square, while when*n*= 1 the domain on integration is a bounded closed interval.**^**Since, as shown in the "*Basic properties*" section, it is exactly a norm.**^**Jones, Peter (1980). "Extension Theorems for BMO".*Indiana University Mathematics Journal*.**29**(1): 41–66. doi:10.1512/iumj.1980.29.29005.**^**See the original paper by Fefferman & Stein (1972), or the paper by Uchiyama (1982) or the comprehensive monograph of Stein (1993, p. 142) for a proof.**^**See the paper Garnett & Jones 1978 for the details.**^**An arc in the unit circle*T*can be defined as the image of a finite interval on the real line**R**under a continuous function whose codomain is*T*itself: a simpler, somewhat naive definition can be found in the entry "Arc (geometry)".**^**See the section on Fefferman theorem of the present entry.**^**See for example Girela, pp. 102–103).**^**See reference Stein 1993, p. 180.**^**Garnett 2007**^**See the reference paper by Garnett & Jones 1982 for a comprehensive development of these themes.- ^
^{a}^{b}See reference Stein 1993, p. 140. **^**See reference Stein 1993, p. 197.

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*Bulletin of the American Mathematical Society*,**9**(3): 267–291, doi:10.1090/S0273-0979-1983-15185-6, MR 0714990, Zbl 0533.73001. A historical paper about the fruitful interaction of elasticity theory and mathematical analysis. - Lennart, Carleson (1981), "BMO – 10 years' development", in Baslev, Erik,
*18th Scandinavian Congress of Mathematicians. Proceedings, 1980*, Progress in Mathematics,**11**, Boston–Basel–Stuttgart: Birkhäuser Verlag, pp. 3–21, ISBN 3-7643-3040-6, MR 0633348, Zbl 0495.46021. - Nirenberg, Louis (1985), "Commentary on [various papers]", in Moser, Jürgen,
*Fritz John: Collected Papers Volume 2*, Contemporary Mathematicians, Boston–Basel–Stuttgart: Birkhäuser Verlag, pp. 703–710, ISBN 0-8176-3265-4, Zbl 0584.01025

- Fefferman, C. (1971), "Characterizations of bounded mean oscillation",
*Bulletin of the American Mathematical Society*,**77**(4): 587–588, doi:10.1090/S0002-9904-1971-12763-5, MR 0280994, Zbl 0229.46051. - Fefferman, C.; Stein, E. M. (1972), "H
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