Bounded mean oscillation

According to {{Harvtxt|Nirenberg|1985|p=703 and p. 707}},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}}. the space of functions of bounded mean oscillation was introduced by {{Harvtxt|John|1961|pp=410–411}} in connection with his studies of mappings from a bounded set $\backslash Omega$ belonging to $\backslash mathbb\{R\}^n$ into $\backslash mathbb\{R\}^n$ 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 {{harvtxt|John|Nirenberg|1961}},The paper {{harv|John|1961}} just precedes the paper {{harv|John|Nirenberg|1961}} in volume 14 of the Communications on Pure and Applied Mathematics. where several properties of this function spaces were proved. The next important step in the development of the theory was the proof by Charles FeffermanElias Stein credits only Fefferman for the discovery of this fact: see {{harv|Stein|1993|p=139}}. of the duality between BMO and the Hardy space $H^1$, 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.See his proof in the paper {{Harvnb|Uchiyama|1982}}.

{{EquationRef|1|Definition 1.}} The mean oscillation of a locally integrable function $u$ over a hypercubeWhen $n\; =\; 3$ or $n\; =\; 2$, $Q$ is respectively a cube or a square, while when $n\; =\; 1$ the domain on integration is a bounded closed interval. $Q$ in $\backslash mathbb\{R\}^n$ is defined as the value of the following integral:

$$\backslash frac\; 1$$

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. $$u\_Q\; =\; \backslash frac\; 1$$

  Q

  \int_Q u(y)\,\mathrm{d}y.

{{EquationRef|2|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 $\backslash mathbb\{R\}^n$, is finite.

Note 1. The supremum of the mean oscillation is called the BMO norm of $u$.Since, as shown in the "Basic properties" section, it is exactly a norm. and is denoted by $\backslash |u\backslash |\_\backslash operatorname\{BMO\}$ (and in some instances it is also denoted $\backslash |\; u\; \backslash |\_*$).

Note 2. The use of cubes $Q$ in $\backslash mathbb\{R\}^n$ as the integration domains on which the {{EquationNote|1|mean oscillation}} is calculated, is not mandatory: {{harvtxt|Wiegerinck|2001}} uses balls instead and, as remarked by {{harvtxt|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 $\backslash Omega$ is $\backslash operatorname\{BMO\}(\backslash Omega)$: when $\backslash Omega\; =\; \backslash mathbb\{R\}^n$, $\backslash operatorname\{BMO\}(\backslash mathbb\{R\}^n)$ is abbreviated as $\backslash operatorname\{BMO\}$.
• The BMO norm of a given BMO function $u$ is denoted by $\backslash |\; u\; \backslash |\_\backslash operatorname\{BMO\}$: in some instances, it is also denoted as $\backslash |\; u\; \backslash |\_*$.

$\backslash operatorname\{BMO\}$ functions are locally $L^p$ if

:$\backslash |u\backslash |\_\backslash text\{BMO\}\backslash simeq\backslash sup\_Q\backslash left(\backslash frac\{1\}$

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

As the name suggests, the mean or average of a function in $\backslash operatorname\{BMO\}$ 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

:$|f\_R-f\_Q|\backslash leq\; C\backslash |f\backslash |\_\backslash text\{BMO\}$

where $C\; >\; 0$ is some universal constant. This property is, in fact, equivalent to $f$ being in $\backslash operatorname\{BMO\}$, that is, if $f$ is a locally integrable function such that $|f\_R-f\_Q|\; \backslash leq\; C$ for all dyadic cubes $Q$ and $R$ adjacent in the sense described above and $f$ is in dyadic $\backslash operatorname\{BMO\}$ (where the supremum is only taken over dyadic cubes $Q$), then $f$ is in $\backslash operatorname\{BMO\}$.{{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}}

{{harvtxt|Fefferman|1971}} showed that the $\backslash operatorname\{BMO\}$ space is dual to $H^1$, the Hardy space with $p\; =\; 1$.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. The pairing between $f\; \backslash in\; H^1$ and $g\; \backslash in\; \backslash operatorname\{BMO\}$ is given by

:$(f,g)\; =\; \backslash int\_\{\backslash R^n\}\; f(x)\; g(x)\; \backslash ,\; \backslash mathrm\{d\}x$

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 each $f\backslash in\backslash operatorname\{BMO\}\backslash left(\backslash mathbb\{R\}^n\backslash right)$, there are constants $c\_1,c\_2>0$ (independent of $f$), such that for any cube $Q$ in $\backslash mathbb\{R\}^n$,

$$\backslash left\; |\; \backslash left\; \backslash \{x\backslash in\; Q:\; |f-f\_\{Q\}|>\backslash lambda\; \backslash right\; \backslash \}\; \backslash right\; |\backslash leq\; c\_\{1\}\backslash exp\; \backslash left\; (-c\_2\; \backslash frac\{\backslash lambda\}\{\backslash |f\backslash |\_\backslash operatorname\{BMO\}\}\; \backslash right\; )|Q|.$$

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

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

:$\backslash sup\_\{Q\backslash subseteq\backslash mathbb\{R\}^n\}\backslash frac\{1\}$

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

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

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 {{EquationNote|2|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

:$\backslash frac\; 1$

i.e. such that its {{EquationNote|1|mean oscillation}} over every arc I of the unit circleAn 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)". is bounded. Here as before f_I 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:

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

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 Rⁿ × (0, ∞].See the section on Fefferman theorem of the present entry. In the theory of Complex and Harmonic analysis on the unit disk, his result is stated as follows.See for example {{harvtxt|Girela|2001|pp=102–103}}. 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 transformation T_g

:$T\_g(f)\; =\; \backslash lim\_\{r\; \backslash to\; 1\}\; \backslash int\_\{-\backslash pi\}^\backslash pi\; \backslash bar\{g\}(e^\{i\backslash theta\})\; f(re^\{i\backslash theta\})\; \backslash ,\; \backslash mathrm\{d\}\backslash theta$

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 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.See reference {{harvnb|Stein|1993|p=180}}.

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

:$f=f\_1\; +\; H\; f\_2\; +\; \backslash alpha$

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

The BMO norm is then equivalent to the infimum of $\backslash |f\_1\backslash |\_\backslash infty\; +\; \backslash |f\_2\backslash |\_\backslash infty$ 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.{{harvnb|Garnett|2007}}

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 ||•||_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 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. In the case of BMO on Tⁿ instead of Rⁿ, a function f is such that ||f(•−x)||_BMOd ≤ 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. T. Mei, BMO is the intersection of two translates of dyadic BMO. C. R. Math. Acad. Sci. Paris 336 (2003), no. 12, 1003-1006.

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.See the referenced paper by {{Harvnb|Garnett|Jones|1982}} for a comprehensive development of these themes.

Examples of $\backslash operatorname\{BMO\}$ functions include the following:

• All bounded (measurable) functions. If $f$ is in $L^\backslash infty$, then $\backslash |\; f\; \backslash |\_\backslash operatorname\{BMO\}\; \backslash leq\; 2\backslash |\; f\; \backslash |\_\backslash infty$:See reference {{harvnb|Stein|1993|p=140}}. however, the converse is not true as the following example shows.
• The function $\backslash log|P|$ for any polynomial $P$ that is not identically zero: in particular, this is true also for $|P(x)|\; =\; |x|$.
• If $w$ is an $A\_\backslash infty$ weight, then $\backslash log(w)$ is $\backslash operatorname\{BMO\}$. Conversely, if $f$ is $BMO$, then $e^\{\backslash delta\; f\}$ is an $A\_\backslash infty$ weight for $\backslash delta>0$ small enough: this fact is a consequence of the John–Nirenberg Inequality.See reference {{harvnb|Stein|1993|p=197}}.

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{{refend}}

Category:Function spaces

Category:Functional analysis

Category:Harmonic analysis