Bounded mean oscillation

Last updated February 17, 2024

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 L^∞ of essentially bounded functions plays in the theory of L^p-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
Definition
Notation
Basic properties
BMO functions are locally p–integrable
BMO is a Banach space
Averages of adjacent cubes are comparable
BMO is the dual vector space of H1
The John–Nirenberg Inequality
Generalizations and extensions
The spaces BMOH and BMOA
The space VMO
Relation to the Hilbert transform
The dyadic BMO space
Examples
Notes
References
Historical references
Scientific references

Historical note

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 Rⁿ into Rⁿ 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 H¹, 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

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

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

where

|Q| is the volume of Q, i.e. its Lebesgue measure
u_Q is the average value of u on the cube Q, i.e. $u_{Q}={\frac {1}{|Q|}}\int _{Q}u(y)\,\mathrm {d} y.$

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 Rⁿ, 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 Rⁿ 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.

Notation

The universally adopted notation used for the set of BMO functions on a given domain $Ω$ is BMO( $Ω$ ): when $Ω$ = Rⁿ, BMO(Rⁿ) is 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||_∗.

Basic properties

BMO functions are locally p–integrable

BMO functions are locally L^p if 0 < p < ∞, but need not be locally bounded. In fact, using the John-Nirenberg Inequality, we can prove that

\|u\|_{\text{BMO}}\simeq \sup _{Q}\left({\frac {1}{|Q|}}\int _{Q}|u-u_{Q}|^{p}dx\right)^{1/p}.

BMO is a Banach space

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 the quotient space of BMO functions modulo the space of constant functions on the domain considered.

Averages of adjacent cubes are comparable

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

|f_{R}-f_{Q}|\leq C\|f\|_{\text{BMO}}

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−f_Q| ≤ C for all dyadic cubes Q and R adjacent in the sense described above and f is in dyadic BMO (where the supremum is only taken over dyadic cubes Q), then f is in BMO.^[7]

BMO is the dual vector space of H¹

Fefferman (1971) showed that the BMO space is dual to H¹, the Hardy space with p = 1.^[8] The pairing between f ∈H¹ and g ∈ BMO is given by

(f,g)=\int _{\mathbb {R} ^{n}}f(x)g(x)\,\mathrm {d} x

though some care is needed in defining this integral, as it does not in general converge absolutely.

The John–Nirenberg Inequality

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.

Statement

For each $f\in \operatorname {BMO} \left(\mathbb {R} ^{n}\right)$ , there are constants $c_{1},c_{2}>0$ (independent of f), such that for any cube $Q$ in $\mathbb {R} ^{n}$ ,

\left|\left\{x\in Q:|f-f_{Q}|>\lambda \right\}\right|\leq c_{1}\exp \left(-c_{2}{\frac {\lambda }{\|f\|_{\text{BMO}}}}\right)|Q|.

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.

A consequence: the distance in BMO to L^∞

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

\sup _{Q\subseteq \mathbb {R} ^{n}}{\frac {1}{|Q|}}\int _{Q}e^{\left|f-f_{Q}\right|/A}\mathrm {d} x<\infty .

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(Rⁿ) the following two-sided inequality holds

{\frac {1}{C}}A(f)\leq \inf _{g\in L^{\infty }}\|f-g\|_{\text{BMO}}\leq CA(f).

Generalizations and extensions

The spaces BMOH and BMOA

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

{\frac {1}{|I|}}\int _{I}|f(y)-f_{I}|\,\mathrm {d} y<C<+\infty

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:

u(a)={\frac {1}{2\pi }}\int _{\mathbf {T} }{\frac {1-|a|^{2}}{\left|a-e^{i\theta }\right|^{2}}}f(e^{i\theta })\,\mathrm {d} \theta

equipped with the norm:

\|u\|_{\text{BMOH}}=\sup _{|a|<1}\left\{{\frac {1}{2\pi }}\int _{\mathbf {T} }{\frac {1-|a|^{2}}{|a-e^{i\theta }|^{2}}}\left|f(e^{i\theta })-u(a)\right|\mathrm {d} \theta \right\}

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

BMOA as the dual space of H¹(D)

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 H^p(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 transformationT_g

T_{g}(f)=\lim _{r\to 1}\int _{-\pi }^{\pi }{\bar {g}}(e^{i\theta })f(re^{i\theta })\,\mathrm {d} \theta

Notice that although the limit always exists for an H¹ function f and T_g 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

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 H¹ is the dual of VMO.^[13]

Relation to the Hilbert transform

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

f=f_{1}+Hf_{2}+\alpha

where f_i ∈ L^∞, α is a constant and H is the Hilbert transform.

The BMO norm is then equivalent to the infimum of $\|f_{1}\|_{\infty }+\|f_{2}\|_{\infty }$ 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 R.^[14]

The dyadic BMO space

Let Δ denote the set of dyadic cubes in Rⁿ. 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 ||•||_{BMO_d}.

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 f is such that ||f(•−x)||_{BMO_d} ≤ C for all x in Rⁿ for some C > 0, then by the one-third trick f is also in BMO. In the case of BMO on Tⁿ instead of Rⁿ, a function f is such that ||f(•−x)||_{BMO_d} ≤ C for n+1 suitably chosen x, then f is also in BMO. This means BMO(Tⁿ ) is the intersection of n+1 translation of dyadic BMO. By duality, H¹(Tⁿ ) is the sum of n+1 translation of dyadic H¹.^[15]

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.^[16]

Examples

Examples of BMO functions include the following:

All bounded (measurable) functions. If f is in L^∞, then ||f||_BMO ≤ 2||f||_∞:^[17] 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|.^[17]
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.^[18]

Notes

↑ 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 (2001 , pp. 102–103).
↑ See reference Stein 1993 , p. 180.
↑ Garnett 2007
↑ T. Mei, BMO is the intersection of two translates of dyadic BMO. C. R. Math. Acad. Sci. Paris 336 (2003), no. 12, 1003-1006.
↑ See the referenced paper by Garnett & Jones 1982 for a comprehensive development of these themes.
1 2 See reference Stein 1993 , p. 140.
↑ See reference Stein 1993 , p. 197.

Related Research Articles

In mathematics, an $n$ -sphere or hypersphere is an $n$ -dimensional generalization of the 1-dimensional circle and 2-dimensional sphere to any non-negative integer $n$ . The $n$ -sphere is the setting for $n$ -dimensional spherical geometry.

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols $, (where is the nabla operator), or . In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian Δ f (p) of a function f at a point p measures by how much the average value of f over small spheres or balls centered at p deviates from f (p) .$

In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution. There are two equivalent parameterizations in common use:

With a shape parameter $k$ and a scale parameter $θ$
With a shape parameter $and an inverse scale parameter, called a rate parameter.$

In complex analysis, the Hardy spacesH^p are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz, who named them after G. H. Hardy, because of the paper. In real analysis Hardy spaces are certain spaces of distributions on the real line, which are boundary values of the holomorphic functions of the complex Hardy spaces, and are related to the L^p spaces of functional analysis. For 1 ≤ p < ∞ these real Hardy spaces H^p are certain subsets of L^p, while for p < 1 the L^p spaces have some undesirable properties, and the Hardy spaces are much better behaved.

In mathematics and signal processing, the Hilbert transform is a specific singular integral that takes a function, $u (t)$ of a real variable and produces another function of a real variable $H(u)(t)$ . The Hilbert transform is given by the Cauchy principal value of the convolution with the function $(see § Definition). The Hilbert transform has a particularly simple representation in the frequency domain: It imparts a phase shift of \pm90° (π /2 radians) to every frequency component of a function, the sign of the shift depending on the sign of the frequency (see § Relationship with the Fourier transform). The Hilbert transform is important in signal processing, where it is a component of the analytic representation of a real-valued signal u (t) . The Hilbert transform was first introduced by David Hilbert in this setting, to solve a special case of the Riemann-Hilbert problem for analytic functions.$

In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of L^p-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function.

In the mathematical discipline of complex analysis, the analytic capacity of a compact subset K of the complex plane is a number that denotes "how big" a bounded analytic function on C \ K can become. Roughly speaking, γ(K) measures the size of the unit ball of the space of bounded analytic functions outside K.

In mathematics, the Riesz–Thorin theorem, often referred to as the Riesz–Thorin interpolation theorem or the Riesz–Thorin convexity theorem, is a result about interpolation of operators. It is named after Marcel Riesz and his student G. Olof Thorin.

Directional statistics is the subdiscipline of statistics that deals with directions, axes or rotations in Rⁿ. More generally, directional statistics deals with observations on compact Riemannian manifolds including the Stiefel manifold.

In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the Rellich–Kondrachov theorem showing that under slightly stronger conditions some Sobolev spaces are compactly embedded in others. They are named after Sergei Lvovich Sobolev.

Stochastic approximation methods are a family of iterative methods typically used for root-finding problems or for optimization problems. The recursive update rules of stochastic approximation methods can be used, among other things, for solving linear systems when the collected data is corrupted by noise, or for approximating extreme values of functions which cannot be computed directly, but only estimated via noisy observations.

In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator

In mathematics, the Dirichlet energy is a measure of how variable a function is. More abstractly, it is a quadratic functional on the Sobolev space $H 1$ . The Dirichlet energy is intimately connected to Laplace's equation and is named after the German mathematician Peter Gustav Lejeune Dirichlet.

In mathematics, the class of Muckenhoupt weights $A p$ consists of those weights $ω$ for which the Hardy–Littlewood maximal operator is bounded on $L p (dω)$ . Specifically, we consider functions $f$ on $R n$ and their associated maximal functions $M (f)$ defined as

In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation, Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed interval to second order differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite. Unlike the classical case, the spectrum may no longer consist of just a countable set of eigenvalues, but may also contain a continuous part. In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure, given by the Titchmarsh–Kodaira formula. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using von Neumann's spectral theorem. It has had important applications in quantum mechanics, operator theory and harmonic analysis on semisimple Lie groups.

In the field of mathematical analysis, an interpolation space is a space which lies "in between" two other Banach spaces. The main applications are in Sobolev spaces, where spaces of functions that have a noninteger number of derivatives are interpolated from the spaces of functions with integer number of derivatives.

In mathematics, quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of a quaternion variable just as functions of a real variable or a complex variable are called.

<span class="mw-page-title-main">Theta model</span>

The theta model, or Ermentrout–Kopell canonical model, is a biological neuron model originally developed to mathematically describe neurons in the animal Aplysia. The model is particularly well-suited to describe neural bursting, which is characterized by periodic transitions between rapid oscillations in the membrane potential followed by quiescence. This bursting behavior is often found in neurons responsible for controlling and maintaining steady rhythms such as breathing, swimming, and digesting. Of the three main classes of bursting neurons, the theta model describes parabolic bursting, which is characterized by a parabolic frequency curve during each burst.

In mathematics, singular integral operators of convolution type are the singular integral operators that arise on Rⁿ and Tⁿ through convolution by distributions; equivalently they are the singular integral operators that commute with translations. The classical examples in harmonic analysis are the harmonic conjugation operator on the circle, the Hilbert transform on the circle and the real line, the Beurling transform in the complex plane and the Riesz transforms in Euclidean space. The continuity of these operators on L² is evident because the Fourier transform converts them into multiplication operators. Continuity on L^p spaces was first established by Marcel Riesz. The classical techniques include the use of Poisson integrals, interpolation theory and the Hardy–Littlewood maximal function. For more general operators, fundamental new techniques, introduced by Alberto Calderón and Antoni Zygmund in 1952, were developed by a number of authors to give general criteria for continuity on L^p spaces. This article explains the theory for the classical operators and sketches the subsequent general theory.

In mathematics, and in particular in mathematical analysis, the Gagliardo–Nirenberg interpolation inequality is a result in the theory of Sobolev spaces that relates the $-norms of different weak derivatives of a function through an interpolation inequality. The theorem is of particular importance in the framework of elliptic partial differential equations and was originally formulated by Emilio Gagliardo and Louis Nirenberg in 1958. The Gagliardo-Nirenberg inequality has found numerous applications in the investigation of nonlinear partial differential equations, and has been generalized to fractional Sobolev spaces by Haim Brezis and Petru Mironescu in the late 2010s.$

References

Historical references

Antman, Stuart (1983), "The influence of elasticity in analysis: modern developments" (PDF), 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 (ed.), 18th Scandinavian Congress of Mathematicians. Proceedings, 1980, Progress in Mathematics, vol. 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 (ed.), Fritz John: Collected Papers Volume 2, Contemporary Mathematicians, Boston–Basel–Stuttgart: Birkhäuser Verlag, pp. 703–710, ISBN 0-8176-3265-4, Zbl 0584.01025

Scientific references

Fefferman, C. (1971), "Characterizations of bounded mean oscillation" (PDF), 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^p spaces of several variables", Acta Mathematica , 129: 137–193, doi: 10.1007/BF02392215 , MR 0447953, Zbl 0257.46078 .
Folland, G.B. (2001) [1994], "Hardy spaces", Encyclopedia of Mathematics , EMS Press .
Garnett, John B. (2007) [1981], Bounded analytic functions, Graduate Texts in Mathematics, vol. 236 (Revised 1st ed.), Springer, pp. xiv+459, ISBN 978-0-387-33621-3, MR 2261424, Zbl 1106.30001 .
Garnett, John. B; Jones, Peter W. (September 1978), "The distance in BMO to L^∞", Annals of Mathematics , Second Series, 108 (2): 373–393, doi:10.2307/1971171, JSTOR 1971171, MR 0506992, Zbl 0358.26010 .
Garnett, John. B; Jones, Peter W. (1982), "BMO from Dyadic BMO", Pacific Journal of Mathematics , 99 (2): 351–371, doi: 10.2140/pjm.1982.99.351 , MR 0658065, Zbl 0516.46021 .
Girela, Daniel (2001), "Analytic functions of bounded mean oscillation", in Aulaskari, Rauno (ed.), Complex function spaces, Proceedings of the summer school, Mekrijärvi, Finland, August 30-September 3, 1999, Univ. Joensuu Dept. Math. Rep. Ser., vol. 4, Joensuu: Joensuu University, Department of Mathematics, pp. 61–170, MR 1820090, Zbl 0981.30026 .
John, F. (1961), "Rotation and strain", Communications on Pure and Applied Mathematics , 14 (3): 391–413, doi:10.1002/cpa.3160140316, MR 0138225, Zbl 0102.17404 .
John, F.; Nirenberg, L. (1961), "On functions of bounded mean oscillation", Communications on Pure and Applied Mathematics , 14 (3): 415–426, doi:10.1002/cpa.3160140317, hdl: 10338.dmlcz/128274 , MR 0131498, Zbl 0102.04302 .
Stein, Elias M. (1993), Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, vol. 43, Princeton, NJ: Princeton University Press, pp. xiv+695, ISBN 0-691-03216-5, MR 1232192, OCLC 27108521, Zbl 0821.42001 .
Uchiyama, Akihito (1982), "A constructive proof of the Fefferman-Stein decomposition of BMO(Rⁿ)", Acta Mathematica , 148: 215–241, doi: 10.1007/BF02392729 , MR 0666111, Zbl 0514.46018 .
Wiegerinck, J. (2001) [1994], "BMO space", Encyclopedia of Mathematics , EMS Press .

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] 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).

[2] The paper ( John 1961 ) just precedes the paper ( John & Nirenberg 1961 ) in volume 14 of the Communications on Pure and Applied Mathematics.

[3] Elias Stein credits only Fefferman for the discovery of this fact: see ( Stein 1993 , p. 139).

[4] See his proof in the paper Uchiyama 1982.

[5] 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.

[6] Since, as shown in the " Basic properties " section, it is exactly a norm.

[Jon80-7] Jones, Peter (1980). "Extension Theorems for BMO". Indiana University Mathematics Journal. 29 (1): 41–66. doi: 10.1512/iumj.1980.29.29005 .

[8] 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.

[GarJon1-9] See the paper Garnett & Jones 1978 for the details.

[10] 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)".

[11] See the section on Fefferman theorem of the present entry.

[12] See for example Girela (2001 , pp. 102–103).

[13] See reference Stein 1993 , p. 180.

[14] Garnett 2007

[Mei-15] T. Mei, BMO is the intersection of two translates of dyadic BMO. C. R. Math. Acad. Sci. Paris 336 (2003), no. 12, 1003-1006.

[GarJon2-16] See the referenced paper by Garnett & Jones 1982 for a comprehensive development of these themes.

[S140-17] 1 2 See reference Stein 1993 , p. 140.

[S197-18] See reference Stein 1993 , p. 197.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]