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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 space]]s ''H<sup>p</sup>'' that the space ''L''<sup>∞</sup> of [[Essential supremum|essentially bounded function]]s plays in the theory of [[Lp space|''L<sup>p</sup>''-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 {{Harvtxt|Nirenberg|1985|p=703 and p. 707}},<ref>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 {{harvtxt|Stein|1993|loc=chapter IV}}.</ref> the space of functions of bounded mean oscillation was introduced by {{Harvtxt|John|1961|pp=410–411}} in connection with his studies of [[Map (mathematics)|mappings]] from a [[bounded set]] <math>\Omega</math> belonging to <math>\mathbb{R}^n</math> into <math>\mathbb{R}^n</math> and the corresponding problems arising from [[Solid mechanics|elasticity theory]], precisely from the concept of [[Deformation (mechanics)#Strain|elastic strain]]: the basic notation was introduced in a closely following paper by {{harvtxt|John|Nirenberg|1961}},<ref>The paper {{harv|John|1961}} just precedes the paper {{harv|John|Nirenberg|1961}} in volume 14 of the [[Communications on Pure and Applied Mathematics]].</ref> 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]]<ref>[[Elias Stein]] credits only Fefferman for the discovery of this fact: see {{harv|Stein|1993|p=139}}.</ref> of the [[Duality (mathematics)|duality]] between '''BMO''' and the [[Hardy space]] <math>H^1</math>, in the noted paper {{Harvnb|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]].<ref>See his proof in the paper {{Harvnb|Uchiyama|1982}}.</ref>

== Definition ==
{{EquationRef|1|Definition 1.}} The '''mean oscillation''' of a [[locally integrable function]] <math>u</math> over a [[hypercube]]<ref>When <math>n = 3</math> or <math>n = 2</math>, <math>Q</math> is respectively a [[cube]] or a [[Square (geometry)|square]], while when <math>n = 1</math> the domain on integration is a [[Interval (mathematics)#Classification of intervals|bounded closed interval]].</ref> <math>Q</math> in <math>\mathbb{R}^n</math> is defined as the value of the following [[integral]]:
<math display="block"> \frac 1 {|Q|} \int_Q |u(y)-u_Q|\,\mathrm{d}y</math>
where
*<math>|Q|</math> is the [[Volume (mathematics)|volume]] of <math>Q</math>, i.e. its [[Lebesgue measure]]
*<math>u_Q</math> is the average value of <math>u</math> on the cube <math>Q</math>, i.e. <math display="block"> u_Q = \frac 1 {|Q|}\int_Q u(y)\,\mathrm{d}y.</math>

{{EquationRef|2|Definition 2.}} A '''BMO function''' is a locally integrable function <math>u</math> whose mean oscillation [[supremum]], taken over the set of all [[cube]]s <math>Q</math> contained in <math>\mathbb{R}^n</math>, is finite.

'''Note 1'''. The supremum of the mean oscillation is called the '''BMO norm''' of <math>u</math>.<ref>Since, as shown in the "''[[Bounded mean oscillation#BMO is a Banach space|Basic properties]]''" section, it is exactly a [[Norm (mathematics)|norm]].</ref> and is denoted by <math>\|u\|_\operatorname{BMO}</math> (and in some instances it is also denoted <math>\| u \|_*</math>).

'''Note 2'''. The use of [[cube]]s <math>Q</math> in <math>\mathbb{R}^n</math> as the [[Integration (mathematics)|integration]] [[Domain (mathematical analysis)|domains]] on which the {{EquationNote|1|mean oscillation}} is calculated, is not mandatory: {{harvtxt|Wiegerinck|2001}} uses [[Ball (mathematics)|balls]] instead and, as remarked by {{harvtxt|Stein|1993|p=140}}, in doing so a perfectly equivalent definition of [[function (mathematics)|functions]] of bounded mean oscillation arises.

===Notation===

*The universally adopted notation used for the set of BMO functions on a given domain <math>\Omega</math> is <math>\operatorname{BMO}(\Omega)</math>: when <math>\Omega = \mathbb{R}^n</math>, <math>\operatorname{BMO}(\mathbb{R}^n)</math> is abbreviated as <math>\operatorname{BMO}</math>.
*The '''BMO norm''' of  a given BMO function <math>u</math> is denoted by <math>\| u \|_\operatorname{BMO}</math>: in some instances, it is also denoted as <math>\| u \|_*</math>.

== Basic properties ==

===BMO functions are locally ''p''–integrable===
<math>\operatorname{BMO}</math> functions are locally <math>L^p</math> if <math>0<p<\infty</math>, but need not be locally bounded. In fact, using the John-Nirenberg Inequality, we can prove that

:<math> \|u\|_\text{BMO}\simeq\sup_Q\left(\frac{1}{|Q|}\int_Q|u-u_Q|^p dx\right)^{1/p}.</math>

=== BMO is a Banach space ===
[[Constant (mathematics)|Constant functions]] have zero mean oscillation, therefore functions differing for a constant <math>c>0</math> can share the same <math>\operatorname{BMO}</math> norm value even if their difference is not zero [[almost everywhere]]. Therefore, the function <math>\| u \|_\operatorname{BMO}</math> is properly a norm on the [[quotient space (linear algebra)|quotient space]] of <math>\operatorname{BMO}</math> functions [[Modulo (jargon)|modulo]] the space of [[constant function]]s on the domain considered.

===Averages of adjacent cubes are comparable===
As the name suggests, the mean or average of a function in <math>\operatorname{BMO}</math> does not oscillate very much when computing it over cubes close to each other in position and scale. Precisely, if <math>Q</math> and <math>R</math> are [[dyadic cubes]] such that their boundaries touch and the side length of <math>Q</math> is no less than one-half the side length of <math>R</math> (and vice versa), then
:<math> |f_R-f_Q|\leq C\|f\|_\text{BMO}</math>

where <math>C > 0</math> is some universal constant. This property is, in fact, equivalent to <math>f</math> being in <math>\operatorname{BMO}</math>, that is, if <math>f</math> is a locally integrable function such that <math>|f_R-f_Q| \leq C</math> for all dyadic cubes <math>Q</math> and <math>R</math> adjacent in the sense described above and <math>f</math> is in dyadic <math>\operatorname{BMO}</math> (where the supremum is only taken over dyadic cubes <math>Q</math>), then <math>f</math> is in <math>\operatorname{BMO}</math>.<ref name="Jon80">{{cite journal|last1=Jones|first1=Peter| title=Extension Theorems for BMO |journal=Indiana University Mathematics Journal |date=1980|volume=29| issue=1 |pages=41–66 |doi=10.1512/iumj.1980.29.29005 |doi-access=free}}</ref>

=== BMO is the dual vector space of ''H''<sup>1</sup> ===
{{harvtxt|Fefferman|1971}} showed that the <math>\operatorname{BMO}</math> space is dual to <math>H^1</math>, the [[Hardy space]] with <math>p = 1</math>.<ref>See the original paper by {{harvtxt|Fefferman|Stein|1972}}, or the paper by {{Harvtxt|Uchiyama|1982}} or the comprehensive [[monograph]] of {{Harvtxt|Stein|1993|p=142}} for a proof.</ref>  The pairing between <math>f \in H^1</math> and <math>g \in \operatorname{BMO}</math> is given by
:<math>(f,g) = \int_{\R^n} f(x) g(x) \, \mathrm{d}x</math>
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 <math>f\in\operatorname{BMO}\left(\mathbb{R}^n\right)</math>, there are constants <math>c_1,c_2>0</math> (independent of <math>f</math>), such that for any cube <math>Q</math> in <math>\mathbb{R}^n</math>,
<math display="block"> \left | \left \{x\in Q: |f-f_{Q}|>\lambda \right \} \right |\leq c_{1}\exp \left (-c_2 \frac{\lambda}{\|f\|_\operatorname{BMO}} \right )|Q|.</math>

Conversely, if this inequality holds over all [[cube]]s with some constant <math>C</math> in place of <math>\| f \|_\operatorname{BMO}</math>, then <math>f</math> is in <math>\operatorname{BMO}</math> with norm at most a constant times <math>C</math>.

====A consequence: the distance in BMO to ''L<sup>∞</sup>''====
The John–Nirenberg inequality can actually give more information than just the <math>\operatorname{BMO}</math> norm of a function. For a locally integrable function <math>f</math>, let <math>A(f)</math> be the infimal <math>A>0</math> for which

:<math>\sup_{Q\subseteq\mathbb{R}^n}\frac{1}{|Q|} \int_{Q}e^{\left|f-f_{Q}\right|/A} \mathrm{d}x<\infty.</math>

The John–Nirenberg inequality implies that <math>A(f)\leq C \| f \|_\operatorname{BMO}</math> for some universal constant <math>C</math>. For an [[Lp space#Lp spaces and Lebesgue integrals|<math>L^\infty</math>]] function, however, the above inequality will hold for all <math>A>0</math>. In other words, <math>A(f)=0</math> if <math>f</math> is in <math>L^\infty</math>. Hence the constant <math>A(f)</math> gives us a way of measuring how far a function in <math>\operatorname{BMO}</math> is from the subspace <math>L^\infty</math>. This statement can be made more precise:<ref name="GarJon1" >See the paper {{Harvnb|Garnett|Jones|1978}} for the details.</ref> there is a constant <math>C</math>, depending only on the [[Dimension (mathematics)|dimension]] <math>n</math>, such that for any function <math>f\in \operatorname{BMO}(\mathbb{R}^n)</math> the following two-sided inequality holds

:<math> \frac 1 C A(f) \leq \inf_{g\in L^\infty} \|f-g\|_\text{BMO}\leq CA(f).</math>

== Generalizations and extensions ==

=== The spaces BMOH and BMOA ===

When the [[Dimension (mathematics)|dimension]] of the ambient space is 1, the space BMO can be seen as a [[linear subspace]] of [[harmonic function]]s on the [[unit disk]] and plays a major role in the theory of [[Hardy spaces]]: by using {{EquationNote|2|definition 2}}, it is possible to define the BMO(''T'') space on the [[unit circle]] as the space of [[Function (mathematics)|functions]] ''f'' : ''T'' → '''R''' such that

:<math> \frac 1 {|I|}\int_I|f(y)-f_I|\,\mathrm{d}y < C <+\infty</math>

i.e. such that its {{EquationNote|1|mean oscillation}} over every arc I of the [[unit circle]]<ref>An arc in the [[unit circle]] ''T'' can be defined as the [[image (mathematics)|image]] of a [[interval (mathematics)|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)]]".</ref> is bounded. Here as before ''f<sub>I</sub>'' is the mean value of f over the arc I.

{{EquationRef|3|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:

:<math> 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</math>

equipped with the norm:

:<math>\|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\}</math>

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

==== BMOA as the dual space of ''H''<sup>1</sup>(''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 (geometry)|half-space]] '''R'''<sup>''n''</sup> × (0, ∞].<ref>See the [[Bounded mean oscillation#BMO is the dual vector space of H1|section on Fefferman theorem]] of the present entry.</ref> In the theory of Complex and Harmonic analysis on the unit disk, his result is stated as follows.<ref>See for example {{harvtxt|Girela|2001|pp=102–103}}.</ref> Let ''H<sup>p</sup>''(''D'') be the Analytic [[Hardy space]]  on the [[unit Disc]]. For ''p''&nbsp;= 1 we identify (''H''<sup>1</sup>)* with BMOA  by pairing ''f'' ∈ ''H''<sup>1</sup>(''D'') and ''g''&nbsp;∈ BMOA using the ''anti-linear transformation'' ''T<sub>g</sub>''

:<math>T_g(f) = \lim_{r \to 1} \int_{-\pi}^\pi \bar{g}(e^{i\theta}) f(re^{i\theta}) \, \mathrm{d}\theta</math>

Notice that although the limit always exists for an ''H''<sup>1</sup> function f and ''T<sub>g</sub>'' is an element of the dual space  (''H''<sup>1</sup>)*, since the transformation is ''anti-linear'', we don't have an isometric isomorphism between (''H''<sup>1</sup>)* 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 ''H''<sup>1</sup> is the dual of VMO.<ref>See reference {{harvnb|Stein|1993|p=180}}.</ref>

===Relation to the Hilbert transform===
A locally integrable function ''f'' on '''R''' is BMO if and only if it can be written as

:<math> f=f_1 + H f_2 + \alpha </math>

where ''f<sub>i</sub>'' ∈ ''L''<sup>∞</sup>, α is a constant and ''H'' is the [[Hilbert transform]].

The BMO norm is then equivalent to the infimum of <math> \|f_1\|_\infty + \|f_2\|_\infty</math> over all such representations.

Similarly ''f'' is VMO if and only if it can be represented in the above form with ''f<sub>i</sub>'' bounded uniformly continuous functions on '''R'''.<ref>{{harvnb|Garnett|2007}}</ref>

=== The dyadic BMO space ===
Let ''Δ'' denote the set of [[dyadic cubes]] in '''R'''<sup>''n''</sup>. The space '''dyadic BMO''', written BMO<sub>d</sub> 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 ||•||<sub>BMO<sub>''d''</sub></sub>.

This space properly contains BMO.  In particular, the function log(''x'')''χ''<sub>[0,∞)</sub> is a function that is in dyadic BMO but not in BMO. However, if a function ''f'' is such that ||''f''(•−''x'')||<sub>BMO<sub>d</sub></sub> ≤ ''C'' for all ''x'' in '''R'''<sup>''n''</sup> for some ''C'' > 0, then by the [[one-third trick]] ''f'' is also in BMO. In the case of BMO on '''T'''<sup>''n''</sup> instead of '''R'''<sup>''n''</sup>, a function ''f'' is such that ||''f''(•−''x'')||<sub>BMO<sub>d</sub></sub> ≤ ''C'' for n+1 suitably chosen ''x'', then ''f'' is also in BMO. This means BMO('''T'''<sup>''n''</sup> ) is the intersection of n+1 translation of dyadic BMO. By duality,  H<sup>1</sup>('''T'''<sup>''n''</sup> ) is the sum of ''n''+1 translation of dyadic  H<sup>1</sup>.<ref name="Mei"> T. Mei,  BMO is the intersection of two translates of dyadic BMO. C. R. Math. Acad. Sci. Paris 336 (2003), no. 12, 1003-1006.</ref>

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.<ref name="GarJon2" >See the referenced paper by {{Harvnb|Garnett|Jones|1982}} for a comprehensive development of these themes.</ref>

==Examples==
Examples of <math>\operatorname{BMO}</math> functions include the following:
* All bounded (measurable) functions. If <math>f</math> is in <math>L^\infty</math>, then <math>\| f \|_\operatorname{BMO} \leq 2\| f \|_\infty</math>:<ref name="S140">See reference {{harvnb|Stein|1993|p=140}}.</ref> however, the converse is not true as the following example shows.
* The function <math>\log|P|</math> for any polynomial <math>P</math> that is not identically zero: in particular, this is true also for <math>|P(x)| = |x|</math>.<ref name="S140"/>
* If <math>w</math> is an [[Muckenhoupt weights|<math>A_\infty</math> weight]], then <math>\log(w)</math> is <math>\operatorname{BMO}</math>. Conversely, if <math>f</math> is <math>BMO</math>, then <math>e^{\delta f}</math> is an <math>A_\infty</math> weight for <math>\delta>0</math> small enough: this fact is a consequence of the [[Bounded mean oscillation#The John–Nirenberg Inequality|John–Nirenberg Inequality]].<ref name="S197">See reference {{harvnb|Stein|1993|p=197}}.</ref>

==Notes==
{{reflist|29em}}

==References==
{{refbegin}}
===Historical references===
*{{Citation
| last = Antman
| first = Stuart
| author-link = Stuart Antman
| title = The influence of elasticity in analysis: modern developments
| journal = [[Bulletin of the American Mathematical Society]]
| volume = 9
| issue = 3
| pages = 267–291
| year = 1983
| doi = 10.1090/S0273-0979-1983-15185-6
| mr = 714990
| zbl = 0533.73001
| doi-access = free
| url = https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-9/issue-3/The-influence-of-elasticity-on-analysis-Modern-developments/bams/1183551288.pdf
}}. A historical paper about the fruitful interaction of [[elasticity theory]] and [[mathematical analysis]].
*{{Citation
 | first = Carleson
 | last = Lennart
 | author-link = Lennart Carleson
 | editor-last = Baslev
 | editor-first = Erik
 | contribution = BMO – 10 years' development
 | title = 18th Scandinavian Congress of Mathematicians. Proceedings, 1980
 | series = Progress in Mathematics
 | volume = 11
 | year = 1981
 | pages = 3–21
 | place = Boston–Basel–Stuttgart
 | publisher = [[Birkhäuser Verlag]]
 | isbn = 3-7643-3040-6
 | mr = 0633348
 | zbl = 0495.46021 
}}.
*{{Citation
| first = Louis
| last = Nirenberg
| author-link = Louis Nirenberg
| editor-last = Moser
| editor-first = Jürgen 
| editor-link = Jürgen Moser
| contribution = Commentary on [various papers]
| title = Fritz John: Collected Papers Volume 2
| series = Contemporary Mathematicians
| year = 1985
| pages = 703–710
| place = Boston–Basel–Stuttgart
| publisher = [[Birkhäuser Verlag]]
| zbl = 0584.01025
| isbn= 0-8176-3265-4
}}

===Scientific references===
{{sfn whitelist|CITEREFWiegerinck2001}}
*{{citation
| first=C. 
| last=Fefferman
| author-link=Charles Fefferman
| title=Characterizations of bounded mean oscillation
| journal= [[Bulletin of the American Mathematical Society]]
| volume= 77  
| year=1971 
| pages= 587–588
| doi=  10.1090/S0002-9904-1971-12763-5 
| mr= 0280994
| zbl= 0229.46051
| issue=4
| doi-access=free
| url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-77/issue-4/Characterizations-of-bounded-mean-oscillation/bams/1183532944.pdf
}}.
*{{citation
| first1=C. 
| last1= Fefferman
| author-link=Charles Fefferman
| first2=E. M.
| last2=Stein 
| author2-link=Elias M. Stein
| title=H<sup>p</sup> spaces of several variables 
| journal=[[Acta Mathematica]] 
| volume= 129  
| year=1972
| pages=137–193 
| doi=10.1007/BF02392215
| mr= 0447953
| zbl= 0257.46078
| doi-access=free
}}.
*{{springer
| id=H/h110090
| title=Hardy spaces
| first=G.B.
| last= Folland
| author-link=Gerald Folland
}}.
*{{citation
 | last=Garnett
 | first= John B.
 | author-link = John B. Garnett
 | title=Bounded analytic functions
 | series=Graduate Texts in Mathematics
 | volume= 236
 | publisher=Springer
 | year=2007
 | orig-year=1981
 | edition= Revised 1st
 | pages=xiv+459
 | isbn=978-0-387-33621-3
 | mr=2261424
 | zbl=1106.30001
}}.
*{{Citation
  | last1 = Garnett
  | first1 = John. B
  | author-link = John B. Garnett
  | last2 = Jones
  | first2 = Peter W.
  | author2-link = Peter Jones (mathematician)
  | title = The distance in BMO to ''L<sup>∞</sup>''
  | journal = [[Annals of Mathematics]] 
  | series = Second Series
  | volume = 108
  | issue = 2
  | pages = 373–393
  |date=September 1978
  | jstor = 1971171
  | doi = 10.2307/1971171
  | mr = 0506992 
  | zbl = 0358.26010
}}.
*{{Citation
  | last1 = Garnett
  | first1 = John. B
  | author-link = John B. Garnett
  | last2 = Jones
  | first2 = Peter W.
  | author2-link = Peter Jones (mathematician)
  | title = BMO from Dyadic BMO
  | journal = [[Pacific Journal of Mathematics]] 
  | volume = 99
  | issue = 2
  | pages = 351–371
  | year = 1982
  | url = http://projecteuclid.org/euclid.pjm/1102734020
  | doi =   10.2140/pjm.1982.99.351
  | mr = 0658065 
  | zbl = 0516.46021
| doi-access = free
  }}.
*{{citation
| last=Girela
| first= Daniel
| author-link=Daniel Girela
| editor-last = Aulaskari
| editor-first = Rauno
| contribution = Analytic functions of bounded mean oscillation
| title= Complex function spaces, Proceedings of the summer school, Mekrijärvi, Finland, August 30-September 3, 1999
| series=Univ. Joensuu Dept. Math. Rep. Ser. 
| volume= 4
| place = [[Joensuu]]
| publisher= Joensuu University, Department of Mathematics
| year=2001
| pages= 61–170
| mr= 1820090
| zbl= 0981.30026
}}.
*{{citation
| first=F. 
| last=John
| author-link=Fritz John
| title= Rotation and strain
| journal=[[Communications on Pure and Applied Mathematics]]
| volume= 14 
| year=1961
| pages= 391–413 
| mr= 0138225
| zbl= 0102.17404
| doi=10.1002/cpa.3160140316
| issue=3 
}}.
*{{citation
| first1=F. 
| last1=John
| author-link=Fritz John
| first2= L.
| last2= Nirenberg
| author2-link=Louis Nirenberg
| title= On functions of bounded mean oscillation
| journal=[[Communications on Pure and Applied Mathematics]]
| volume= 14 
| year=1961
| pages= 415–426 
| mr = 131498
| zbl= 0102.04302
| doi=10.1002/cpa.3160140317
| issue=3
| hdl=10338.dmlcz/128274
| hdl-access=free
}}.
*{{Citation
| last=Stein
| first= Elias M.
| author-link=Elias M. Stein
| title=Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals 
| series=Princeton Mathematical Series 
| volume= 43
| publisher= [[Princeton University Press]]
| location= Princeton, NJ 
| year=1993
| pages= xiv+695
| url=https://books.google.com/books?id=ljcOSMK7t0EC
| isbn= 0-691-03216-5|oclc=27108521 
| mr= 1232192
| zbl= 0821.42001
}}.
*{{Citation
| last = Uchiyama 
| first = Akihito
| title = A constructive proof of the Fefferman-Stein decomposition of BMO('''R'''<sup>''n''</sup>)
| journal = [[Acta Mathematica]]
| volume = 148
| pages = 215–241
| year = 1982
| doi = 10.1007/BF02392729
| mr = 0666111
| zbl = 0514.46018
| doi-access = free
}}.
*{{springer
| id=b/b110660
| title= BMO space
| first=J.
| last= Wiegerinck}}.
{{refend}}

[[Category:Function spaces]]
[[Category:Functional analysis]]
[[Category:Harmonic analysis]]