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==Functions in Hardy spaces and in $\operatorname{BMO}$.==
Let $D = \{ z \in \mathbf{C} : | z | < 1 \}$ be the unit disc and let, for $1 \leq p < \infty$, $H ^ { p }$ denote the space of holomorphic functions on $D$ (cf. also [[Analytic function|Analytic function]]) for which the supremum

\begin{equation*} \| f \| _ { H ^ { p } } ^ { p } : = \frac { 1 } { 2 \pi } \operatorname { sup } _ { r < 1 } \int _ { - \pi } ^ { \pi } | f ( r e ^ { i \vartheta } ) | ^ { p } d \vartheta \end{equation*}

is finite. If a function $f$ belongs to $H ^ { p }$, $p \geq 1$, then there exists a function $f \in L ^ { p } ( \partial D , d \vartheta / ( 2 \pi ) )$ such that

\begin{equation*} f ( z ) = \int k _ { \vartheta } ( z ) f \left( e ^ { i \vartheta } \right) \frac { d \vartheta } { 2 \pi }. \end{equation*}

Here, the function

\begin{equation*} e ^ { i \vartheta } \mapsto k _ { \vartheta } ( z ) = \frac { 1 - | z | ^ { 2 } } { | z - e ^ { i \vartheta } | ^ { 2 } } \end{equation*}

is the probability density (cf. also [[Density of a probability distribution|Density of a probability distribution]]) of a [[Brownian motion|Brownian motion]] starting at $z \in D$ and exiting $D$ at $e ^ { i \vartheta }$. It is the Poisson kernel (cf. also [[Poisson integral|Poisson integral]]) for the unit disc. A function $\varphi$, defined on $[ - \pi , \pi ]$, belongs to $\operatorname{BMO}$ if there exists a constant $c$ such that $\int _ { I } | \varphi - \varphi _ { I } | ^ { 2 } d \vartheta \leq c ^ { 2 } | I |$, for all intervals $I$ (cf. also [[BMO-space|$\operatorname{BMO}$-space]]). Here, $\varphi _ { I } = \int _ { I } \varphi d \vartheta / | I |$ and $| I |$ denotes the [[Lebesgue measure|Lebesgue measure]] of the interval $I$. Let $\varphi _ { 1 }$ and $\varphi_2$ be bounded real-valued functions defined on the boundary $\partial D$ of $D$, and let $\tilde { \varphi }_{2}$ be the boundary function of the harmonic conjugate function of the harmonic extension to $D$ of $\varphi_2$ (cf. also [[Conjugate harmonic functions|Conjugate harmonic functions]]), so that $\varphi _ { 2 } + i \widetilde { \varphi } _ { 2 }$ is the boundary function of a function which is holomorphic on $D$. Then the function $\varphi _ { 1 } + \tilde { \varphi } _ { 2 }$ belongs to $\operatorname{BMO}$: see [[#References|[a4]]], p. 200, or [[#References|[a9]]], p. 295. The function

\begin{equation*} \varphi ( \vartheta ) := \left| \operatorname { log } \left| \operatorname { tan } \frac { 1 } { 2 } \vartheta \right| \right| \end{equation*}

belongs to $\operatorname{BMO}$, but is not bounded; see [[#References|[a6]]], Chap. VI. Composition with the [[Biholomorphic mapping|biholomorphic mapping]]

\begin{equation*} w \mapsto i \frac { 1 - w } { 1 + w } \end{equation*}

turns $\operatorname{BMO}$-functions of the line into $\operatorname{BMO}$-functions of the circle; see [[#References|[a6]]], p. 226.

==Martingales in Hardy spaces and in $\mathcal{BMO}$.==
Let $B _ { t }$, $t \geq 0$, be Brownian motion starting at $0$ and let $\mathcal{F}$ be the filtration generated by Brownian motion (cf. also [[Stochastic processes, filtering of|Stochastic processes, filtering of]]). Notice that $B _ { t }$, $t \geq 0$, is a continuous [[Gaussian process|Gaussian process]] with covariance $\mathsf{E} B _ { s } B _ { t } = \operatorname { min } ( s , t )$. Define, for $0 < p < \infty$, the space of local martingales $\mathcal{M} ^ { p }$ by

<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002053.png"/></td> </tr></table>

Here, $X ^ { * } = \operatorname { sup } _ { t \geq 0 } | X _ { t } |$. Since the martingales are $\mathcal{F}$-martingales, they can be written in the form of an Itô integral:

\begin{equation*} X _ { t } = X _ { 0 } + \int _ { 0 } ^ { t } H _ { s } \cdot d B _ { s }. \end{equation*}

Here, $H$ is a [[Predictable random process|predictable random process]]. Let $A$ be a $( 2 \times 2 )$-matrix, and define the $A$-transform of $X$ by $( A ^ { * } X ) _ { t } = \int _ { 0 } ^ { t } A H _ { s } . d B _ { s }$. Then the [[Martingale|martingale]] $X$ belongs to $\mathcal{M} ^ { 1 }$ if and only all transformed martingales $A  *  X$ have the property that

\begin{equation*} \operatorname { sup } _ { t > 0 } \mathsf{E} [ | ( A ^ { * } X ) _ { t } | ] \end{equation*}

is finite; this is Janson's theorem [[#References|[a8]]]. A martingale $A \in {\cal{M}} ^ { 1 }$ is called an atom if there exists a [[Stopping time|stopping time]] $T$ such that

i) $A _ { t } = 0$ if $t \leq T$; and

ii)

\begin{equation*} A ^ { * } = \operatorname { sup } _ { t \geq 0 } | A _ { t } | \leq \frac { 1 } { \mathsf{P} [ T < \infty ] }. \end{equation*}

Since for atoms $A ^ { * } = 0$ on the event $\{ T = \infty \}$, it follows that $\| A \| _ { 1 } = \mathsf{E} [ A ^ { * } ]$. Moreover, every $X \in \mathcal{M} ^ { 1 }$ can be viewed as a limit of the form

\begin{equation*} X = \mathcal{M} ^ { 1 } - \operatorname { lim } _ { N \rightarrow \infty } \sum _ { n = - N } ^ { n = N } c _ { n } A ^ { n }, \end{equation*}

where every $A ^ { n }$ is an atom and where $\sum _ { x \in \mathbf{N} } |c_n| \|X\|_1$. A local martingale $Y$ is said to have to bounded mean oscillation (notation $Y \in \mathcal{BMO}$) if there exists a constant $c$ such that

\begin{equation*} \mathsf{E} | Y _ { \infty } - Y _ { T } | \leq c \mathsf{P} [ T < \infty ] \end{equation*}

for all $\mathcal{F}$-stopping times $T$. The infimum of the constants $c$ is the $\mathcal{BMO}$-norm of $Y$. It is denoted by $\| Y \| _{*}$. The above inequality is equivalent to

\begin{equation*} \mathsf{E} \left[ | Y _ { \infty } - Y _ { T } | | \mathcal{F} _ { T } \right] \leq c \ \text{almost surely}. \end{equation*}

Let $X$ be a non-negative martingale. Put $X ^ { * } = \operatorname { sup } _ { s \geq 0 } X _ { s }$. Then $X$ belongs to $\mathcal{M} ^ { 1 }$ if and only if $\mathsf{E} [ X _ { \infty } \operatorname { log } ^ { + } X _ { \infty } ]$ is finite. More precisely, the following inequalities are valid:

\begin{equation*} \mathsf{E} [ X _ { 0 } ] + \mathsf{E} \left[ X _ { \infty } \operatorname { log }^+ \frac { X _ { \infty } } { \mathsf{E} [ X _ { 0 } ] } \right] \leq \end{equation*}

\begin{equation*} \leq \mathsf{E} [ X ^ { * } ] \leq \end{equation*}

\begin{equation*} \leq 2 \mathsf{E} [ X _ { 0 } ] + 2 \mathsf{E} \left[ X _ { \infty } \operatorname { log } ^{+} \frac { X _ { \infty } } { \mathsf{E} [ X _ { 0 } ] } \right]. \end{equation*}

For details, see e.g. [[#References|[a4]]], p. 149. Let $Y _ { t } = B _ { \operatorname { min }  ( t , 1 )}$. Then $Y$ is an unbounded martingale in $\mathcal{BMO}$. Two main versions of the John–Nirenberg inequalities are as follows.

===Analytic version of the John–Nirenberg inequality.===
There are constants $C$, $\gamma \in ( 0 , \infty )$, such that, for any function $\varphi \in \operatorname{BMO}$ for which $\| \varphi \| _ { * } \leq 1$, the inequality

\begin{equation*} | \{ \vartheta \in I : | \varphi ( e ^ { i \vartheta } ) - \varphi _ { I } | \geq \lambda \} | \leq C e ^ { - \gamma \lambda } | I | \end{equation*}

is valid for all intervals $I \subset [ - \pi , \pi ]$.

===Probabilistic version of the John–Nirenberg inequality.===
There exists a constant $C$ such that for any martingale $X \in \mathcal{M} ^ { 1 }$ for which $\| X \|  { * } \leq 1$, the inequality $\mathsf{P} [ X ^ { * } > \lambda ] \leq C e ^ { - \lambda / e }$ is valid. For the same constant $C$, the inequality

\begin{equation*} \mathsf{P} \left[ \operatorname { sup } _ { t \geq T } | X _ { t } - X _ { T } | > \lambda \right] \leq C  e^ { - \lambda / e } \mathsf{P} [ T < \infty ] \end{equation*}

is valid for all $\mathcal{F}$-stopping times $T$ and for all $X \in \mathcal{M} ^ { 1 }$ for which $\| X \|  { * } \leq 1$.

As a consequence, for $\varphi \in \operatorname{BMO}$ integrals of the form $\int _ { \partial D } \operatorname { exp } \left( \varepsilon | \varphi ( e ^ { i \vartheta } ) - \varphi _ { I } | \right) d \vartheta$ are finite for $\varepsilon > 0$ sufficiently small.

==Duality between $H ^ { 1 }$ and $\operatorname{BMO}$.==
The John–Nirenberg inequalities can be employed to prove the duality between the space of holomorphic functions $H _ { 0 } ^ { 1 }$ and $\operatorname{BMO}$ and between $\mathcal{M} ^ { 1 }$ and $\mathcal{BMO}$.

===Duality between $H _ { 0 } ^ { 1 }$ and $\operatorname{BMO}$ (analytic version).===
The duality between $H _ { 0 } ^ { 1 } = \{ f \in H ^ { 1 } : f ( 0 ) = 0 \}$ and $\operatorname{BMO}$ is given by

\begin{equation*} ( f , h ) \mapsto \int _ { \partial D } u ( e ^ { i \vartheta } ) h ( e ^ { i \vartheta } ) \frac { d \vartheta } { 2 \pi }, \end{equation*}

where $u ( e ^ { i \vartheta } ) = \operatorname { lim } _ { r \uparrow 1 } \operatorname { Re } f ( r e ^ { i \vartheta } )$ ($f \in H _ { 0 } ^ { 1 }$, $h \in \operatorname{BMO}$).

===Duality between $\mathcal{M} ^ { 1 }$ and $\mathcal{BMO}$ (probabilistic version).===
Let $X$ be a martingale in $\mathcal{M} ^ { 1 }$ and let $Y$ be a martingale in $\mathcal{BMO}$. The duality between these martingales is given by $\mathsf{E} [ X _ { \infty } Y _ { \infty } ]$. Here, $X _ { \infty } = \operatorname { lim } _ { t \rightarrow \infty } X _ { t }$ and $Y _ { \infty } = \operatorname { lim } _ { t \rightarrow \infty } Y _ { t }$.

There exists a more or less canonical way to identify holomorphic functions in $H ^ { 1 }$ and certain continuous martingales in $\mathcal{M} ^ { 1 }$. Moreover, the same is true for functions of bounded mean oscillation (functions in $\operatorname{BMO}$) and certain continuous martingales in $\mathcal{BMO}$. Consequently, the duality between $H ^ { 1 }$ and $\operatorname{BMO}$ can also be extended to a duality between $\mathcal{M} ^ { 1 }$-martingales and $\mathcal{BMO}$-martingales.

The relationship between $H ^ { 1 }$ (respectively, $\operatorname{BMO}$) and a closed subspace of $\mathcal{M} ^ { 1 }$ (respectively, $\mathcal{BMO}$) is determined via the following equalities. For $f \in H ^ { 1 }$ one writes $u = \operatorname { Re } f$ and $U _ { t } = u ( B _ { \operatorname { min }  ( t , \tau )} )$, and for $h \in \operatorname{BMO}$ one writes $H _ { t } = h ( B _ { \operatorname { min }  ( t , \tau )} )$, where, as above, $B _ { t }$ is two-dimensional Brownian motion starting at $0$, and where $\tau = \operatorname { inf } \{ t > 0 : | B _ { t } | = 1 \}$. Then the martingale $U$ belongs to $\mathcal{M} ^ { 1 }$, and $H$ is a member of $\mathcal{BMO}$. The fact that $H ^ { 1 }$ can be considered as a closed subspace of $\mathcal{M} ^ { 1 }$ is a consequence of the following

\begin{equation*} c \mathsf{E} \left[ \left| U _ { \tau } ^ { * } \right| ^ { p } \right] \leq \operatorname { sup } _ { 0 < r < 1 } \int _ { \partial D } | f ( r e ^ { i \vartheta } ) | ^ { p } \frac { d \vartheta } { 2 \pi } \leq C \mathsf{E} \left[ \left| U _ { \tau } ^ { * } \right| ^ { p } \right], \end{equation*}

$f \in H _ { 0 } ^ { p }$, $U _ { t } = \operatorname { Re } f ( B _ { t } )$, $U _ { \tau } ^ { * } = \operatorname { sup } _ { 0  \leq t < \tau} | U _ { t } |$.

An important equality in the proof of these dualities is the following result: Let $f _ { 1 } = u _ { 1 } + i v _ { 1 }$ and $f _ { 2 } = u _ { 2 } + i v _ { 2 }$ be functions in $H _ { 0 } ^ { 2 }$. Then

\begin{equation*} \mathsf{E} [ U _ { \infty } ^ { 1 } U _ { \infty } ^ { 2 } ] = \int _ { \partial D } u _ { 1 } u _ { 2 } \frac { d \vartheta } { 2 \pi } = \int _ { \partial D } v _ { 1 } v _ { 2 } \frac { d \vartheta } { 2 \pi } = \mathsf{E} [ V _ { \infty } ^ { 1 } V _ { \infty } ^ { 2 } ]. \end{equation*}

Here, $U _ { t } ^ { j } = u _ { j } ( B _ { \operatorname { min }( t , \tau ) }  )$, $j = 1,2$. A similar convention is used for $V _ { t } ^ { j }$, $j = 1,2$. In the first (and in the final) equality, the distribution of $\tau$ is used: $\mathsf {P} [ \tau \in I ] = | I | / ( 2 \pi )$. The other equalities depend on the fact that a process like $U _ { t } ^ { 1 } U _ { t } ^ { 2 } - \int _ { 0 } ^ { t } \nabla u _ { 1 } ( B _ { s } ) . \nabla u _ { 2 } ( B _ { s } ) d s$ is a martingale, which follows from Itô calculus in conjunction with the harmonicity of the functions $u_1$ and $u_2$. Next, let $\varphi$ be a function in $\operatorname{BMO}$. Denote by $h$ the harmonic extension of $\varphi$ to $D$. Put $Y _ { t } = h ( B _ { \operatorname { min }  ( t , \tau )} )$. Then $Y _ { t }$ is a continuous martingale. Let $T$ be any stopping time. From the Markov property it follows that $\mathsf{E} \left[ | Y _ { \infty } - Y _ { T } | ^ { 2 } | \mathcal{F} _ { T } \right] = w ( B _ { \operatorname { min }  ( T , \tau )} )$, where

\begin{equation*} w ( z ) = \int k _ { \vartheta } ( z ) | \varphi ( e ^ { i \vartheta } ) - h ( z ) | ^ { 2 } \frac { d \vartheta } { 2 \pi }, \end{equation*}

with

\begin{equation*} k _ { \vartheta } ( z ) = \frac { 1 - | z | ^ { 2 } } { \left| z - e ^ { i \vartheta }\right|^ 2 }. \end{equation*}

As above, the Poisson kernel for the unit disc $e ^ { i \vartheta } \mapsto k _ { \vartheta } ( z )$ can be viewed as the probability density of a Brownian motion starting at $z \in D$ and exiting $D$ at $e ^ { i \vartheta }$. Since the inequality $w ( z ) \leq c ^ { 2 }$ is equivalent to the inequality

\begin{equation*} \int _ { I } | \varphi - \varphi _ { I } | ^ { 2 } \frac { d \vartheta } { 2 \pi } \leq c _ { 1 } ^ { 2 } | I |, \end{equation*}

for some constant $c _ { 1 } = c _ { 1 } ( c )$, it follows that $\operatorname{BMO}$ can be considered as a closed subspace of $\mathcal{BMO}$: see [[#References|[a6]]], Corol. 2.4; p. 234.

The analytic John–Nirenberg inequality can be viewed as a consequence of a result due to A.P. Calderón and A. Zygmund. Let $u$ be function in $L ^ { 1 } ( I )$ ($I$ is some interval). Suppose $| I | \alpha > \int _ { I } | u ( \vartheta ) | d \vartheta$. Then there exists a pairwise disjoint sequence $\{ I_j \}$ of open subintervals of $I$ such that $| u | \leq \alpha$ almost everywhere on $I \backslash \cup I_{j}$,

\begin{equation*} \alpha \leq \frac { 1 } { | I _ { j } | } \int _ { I _ { j } } | u ( \vartheta ) | d \vartheta < 2 \alpha, \end{equation*}

and

\begin{equation*} \sum | I _ { j } | \leq \frac { 1 } { \alpha } \int _ { I } | u ( \vartheta ) | d \vartheta. \end{equation*}

In [[#References|[a1]]], [[#References|[a6]]], [[#References|[a7]]] and [[#References|[a10]]], extensions of the above can be found. In particular, some of the concepts can be extended to other domains in $\mathbf{C}$ (see [[#References|[a6]]]), in $\mathbf{R} ^ { d }$ and in more general Riemannian manifolds ([[#References|[a1]]], [[#References|[a2]]], [[#References|[a7]]], [[#References|[a10]]]). For a relationship with Carleson measures, see [[#References|[a6]]], Chap. 6. A measure $\lambda$ on $D$ is called a Carleson measure if $\lambda ( S ) \leq K. h$ for some constant $K$ and for all circle sectors $S = \left\{ r e ^ { i \vartheta } : 1 - h \leq r < 1 , | \vartheta - \vartheta _ { 0 } | \leq h \right\}$. A function $\varphi$ belongs to $\operatorname{BMO}$ if and only if

\begin{equation*} | \nabla u ( z ) | ^ { 2 } \operatorname { log } \frac { 1 } { | z | } d x d y \end{equation*}

is a Carleson measure. Here, $u$ is the harmonic extension of $\varphi$. For some other phenomena and related inequalities, see e.g. [[#References|[a3]]], [[#References|[a10]]], and [[#References|[a11]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  M. Biroli,   U. Mosco,   "Sobolev inequalities on homogeneous spaces: Potential theory and degenerate partial differential operators (Parma)"  ''Potential Anal.'' , '''4'''  (1995)  pp. 311–324</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  S.Y.A. Chang,   R. Fefferman,   "A continuous version of duality of $H ^ { 1 }$ with $\operatorname{BMO}$ on the bidisc"  ''Ann. of Math. (2)'' , '''112'''  (1980)  pp. 179–201</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  L. Chevalier,   "Quelles sont les fonctions qui opèrent de $\operatorname{BMO}$ dans $\operatorname{BMO}$ ou de $\operatorname{BMO}$ dans $\overline{L^\infty}$"  ''Bull. London Math. Soc.'' , '''27''' :  6  (1995)  pp. 590–594</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  R. Durrett,   "Brownian motion and martingales in analysis" , Wadsworth  (1984)  (Contains Mathematica analysis and stochastic processes)  {{MR|0750829}} {{ZBL|0554.60075}} </td></tr><tr><td valign="top">[a5]</td> <td valign="top">  J.B. Garnett,   "Two constructions in $B M O$"  G. Weiss (ed.)  S. Wainger (ed.) , ''Harmonic analysis in Euclidean spaces'' , ''Proc. Symp. Pure Math.'' , '''XXXV:1''' , Amer. Math. Soc.  (1979)  pp. 295–301</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  J. Garnett,   "Bounded analytic functions" , Acad. Press  (1981)  {{MR|0628971}} {{ZBL|0469.30024}} </td></tr><tr><td valign="top">[a7]</td> <td valign="top">  R. Hurri-Syrjanen,   "The John–Nirenberg inequality and a Sobolev inequality in general domains"  ''J. Math. Anal. Appl.'' , '''175''' :  2  (1993)  pp. 579–587</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  S. Janson,   "Characterization of $H ^ { 1 }$ by singular integral transformations on martingales and ${\bf R} ^ { n }$"  ''Math. Scand.'' , '''41'''  (1977)  pp. 140–152</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  P. Koosis,   "Introduction to $H ^ { p }$-spaces: with an appendix on Wolff's proof of the corona theorem" , ''London Math. Soc. Lecture Notes'' , '''40''' , London Math. Soc.  (1980)</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  Jia-Yu Li,   "On the Harnack inequality for harmonic functions on complete Riemannian manifolds"  ''Chinese Ann. Math. Ser. B'' , '''14''' :  1  (1993)  pp. 1–12</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  F.J. Martin–Reyes,   A. de la Torre,   "One-sided $B M O$ spaces"  ''J. London Math. Soc. (2)'' , '''49''' :  3  (1994)  pp. 529–542</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  G. Weiss,   "Weak-type inequalities for $H ^ { p }$ and $\operatorname{BMO}$"  G. Weiss (ed.)  S. Wainger (ed.) , ''Harmonic Analysis in Euclidean Spaces'' , ''Proc. Symp. Pure Math.'' , '''XXXV:1''' , Amer. Math. Soc.  (1979)  pp. 295–301</td></tr></table>