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==Functions in Hardy spaces and in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j1200201.png" />.==
 
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j1200202.png" /> be the unit disc and let, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j1200203.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j1200205.png" /> denote the space of holomorphic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j1200206.png" /> (cf. also [[Analytic function|Analytic function]]) for which the supremum
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j1200207.png" /></td> </tr></table>
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If the TeX and formula formatting is correct and if all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
  
is finite. If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j1200208.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j1200209.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002010.png" />, then there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002011.png" /> such that
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002012.png" /></td> </tr></table>
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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
 
Here, the function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002013.png" /></td> </tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002014.png" /> and exiting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002015.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002016.png" />. It is the Poisson kernel (cf. also [[Poisson integral|Poisson integral]]) for the unit disc. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002017.png" />, defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002018.png" />, belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002019.png" /> if there exists a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002020.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002021.png" />, for all intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002022.png" /> (cf. also [[BMO-space|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002023.png" />-space]]). Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002025.png" /> denotes the [[Lebesgue measure|Lebesgue measure]] of the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002026.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002028.png" /> be bounded real-valued functions defined on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002029.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002030.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002031.png" /> be the boundary function of the harmonic conjugate function of the harmonic extension to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002032.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002033.png" /> (cf. also [[Conjugate harmonic functions|Conjugate harmonic functions]]), so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002034.png" /> is the boundary function of a function which is holomorphic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002035.png" />. Then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002036.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002037.png" />: see [[#References|[a4]]], p. 200, or [[#References|[a9]]], p. 295. The function
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002038.png" /></td> </tr></table>
+
\begin{equation*} \varphi ( \vartheta ) := \left| \operatorname { log } \left| \operatorname { tan } \frac { 1 } { 2 } \vartheta \right| \right| \end{equation*}
  
belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002039.png" />, but is not bounded; see [[#References|[a6]]], Chap. VI. Composition with the [[Biholomorphic mapping|biholomorphic mapping]]
+
belongs to $\operatorname{BMO}$, but is not bounded; see [[#References|[a6]]], Chap. VI. Composition with the [[Biholomorphic mapping|biholomorphic mapping]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002040.png" /></td> </tr></table>
+
\begin{equation*} w \mapsto i \frac { 1 - w } { 1 + w } \end{equation*}
  
turns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002041.png" />-functions of the line into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002042.png" />-functions of the circle; see [[#References|[a6]]], p. 226.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002043.png" />.==
+
==Martingales in Hardy spaces and in $\mathcal{BMO}$.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002045.png" />, be Brownian motion starting at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002046.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002047.png" /> be the filtration generated by Brownian motion (cf. also [[Stochastic processes, filtering of|Stochastic processes, filtering of]]). Notice that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002049.png" />, is a continuous [[Gaussian process|Gaussian process]] with covariance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002050.png" />. Define, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002051.png" />, the space of local martingales <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002052.png" /> by
+
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 valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002053.png" /></td> </tr></table>
+
$$
 +
\mathcal{M}^p = \left\{
 +
X : \begin{array}{c}
 +
X \text{ a local martingale with respect to } \mathcal{F}, \\
 +
\mathsf{E} |X^*|^P < \infty
 +
\end{array}
 +
\right\}.
 +
$$
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002054.png" />. Since the martingales are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002056.png" />-martingales, they can be written in the form of an Itô integral:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002057.png" /></td> </tr></table>
+
\begin{equation*} X _ { t } = X _ { 0 } + \int _ { 0 } ^ { t } H _ { s } \cdot d B _ { s }. \end{equation*}
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002058.png" /> is a [[Predictable random process|predictable random process]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002059.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002060.png" />-matrix, and define the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002061.png" />-transform of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002062.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002063.png" />. Then the [[Martingale|martingale]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002064.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002065.png" /> if and only all transformed martingales <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002066.png" /> have the property that
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002067.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { sup } _ { t > 0 } \mathsf{E} [ | ( A ^ { * } X ) _ { t } | ] \end{equation*}
  
is finite; this is Janson's theorem [[#References|[a8]]]. A martingale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002068.png" /> is called an atom if there exists a [[Stopping time|stopping time]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002069.png" /> such that
+
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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002070.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002071.png" />; and
+
i) $A _ { t } = 0$ if $t \leq T$; and
  
 
ii)
 
ii)
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002072.png" /></td> </tr></table>
+
\begin{equation*} A ^ { * } = \operatorname { sup } _ { t \geq 0 } | A _ { t } | \leq \frac { 1 } { \mathsf{P} [ T < \infty ] }. \end{equation*}
  
Since for atoms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002073.png" /> on the event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002074.png" />, it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002075.png" />. Moreover, every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002076.png" /> can be viewed as a limit of the form
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002077.png" /></td> </tr></table>
+
\begin{equation*} X = \mathcal{M} ^ { 1 } - \operatorname { lim } _ { N \rightarrow \infty } \sum _ { n = - N } ^ { n = N } c _ { n } A ^ { n }, \end{equation*}
  
where every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002078.png" /> is an atom and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002079.png" />. A local martingale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002080.png" /> is said to have to bounded mean oscillation (notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002081.png" />) if there exists a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002082.png" /> such that
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002083.png" /></td> </tr></table>
+
\begin{equation*} \mathsf{E} | Y _ { \infty } - Y _ { T } | \leq c \mathsf{P} [ T < \infty ] \end{equation*}
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002084.png" />-stopping times <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002085.png" />. The infimum of the constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002086.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002087.png" />-norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002088.png" />. It is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002089.png" />. The above inequality is equivalent to
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002090.png" /></td> </tr></table>
+
\begin{equation*} \mathsf{E} \left[ | Y _ { \infty } - Y _ { T } | | \mathcal{F} _ { T } \right] \leq c \ \text{almost surely}. \end{equation*}
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002091.png" /> be a non-negative martingale. Put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002092.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002093.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002094.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002095.png" /> is finite. More precisely, the following inequalities are valid:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002096.png" /></td> </tr></table>
+
\begin{equation*} \mathsf{E} [ X _ { 0 } ] + \mathsf{E} \left[ X _ { \infty } \operatorname { log }^+ \frac { X _ { \infty } } { \mathsf{E} [ X _ { 0 } ] } \right] \leq \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002097.png" /></td> </tr></table>
+
\begin{equation*} \leq \mathsf{E} [ X ^ { * } ] \leq \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002098.png" /></td> </tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002099.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020100.png" /> is an unbounded martingale in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020101.png" />. Two main versions of the John–Nirenberg inequalities are as follows.
+
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.===
 
===Analytic version of the John–Nirenberg inequality.===
There are constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020103.png" />, such that, for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020104.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020105.png" />, the 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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020106.png" /></td> </tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020107.png" />.
+
is valid for all intervals $I \subset [ - \pi , \pi ]$.
  
 
===Probabilistic version of the John–Nirenberg inequality.===
 
===Probabilistic version of the John–Nirenberg inequality.===
There exists a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020108.png" /> such that for any martingale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020109.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020110.png" />, the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020111.png" /> is valid. For the same constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020112.png" />, the 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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020113.png" /></td> </tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020114.png" />-stopping times <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020115.png" /> and for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020116.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020117.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020118.png" /> integrals of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020119.png" /> are finite for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020120.png" /> sufficiently small.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020121.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020122.png" />.==
+
==Duality between $H ^ { 1 }$ and $\operatorname{BMO}$.==
The John–Nirenberg inequalities can be employed to prove the duality between the space of holomorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020123.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020124.png" /> and between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020125.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020126.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020127.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020128.png" /> (analytic version).===
+
===Duality between $H _ { 0 } ^ { 1 }$ and $\operatorname{BMO}$ (analytic version).===
The duality between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020129.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020130.png" /> is given by
+
The duality between $H _ { 0 } ^ { 1 } = \{ f \in H ^ { 1 } : f ( 0 ) = 0 \}$ and $\operatorname{BMO}$ is given by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020131.png" /></td> </tr></table>
+
\begin{equation*} ( f , h ) \mapsto \int _ { \partial D } u ( e ^ { i \vartheta } ) h ( e ^ { i \vartheta } ) \frac { d \vartheta } { 2 \pi }, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020132.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020133.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020134.png" />).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020135.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020136.png" /> (probabilistic version).===
+
===Duality between $\mathcal{M} ^ { 1 }$ and $\mathcal{BMO}$ (probabilistic version).===
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020137.png" /> be a martingale in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020138.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020139.png" /> be a martingale in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020140.png" />. The duality between these martingales is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020141.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020142.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020143.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020144.png" /> and certain continuous martingales in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020145.png" />. Moreover, the same is true for functions of bounded mean oscillation (functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020146.png" />) and certain continuous martingales in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020147.png" />. Consequently, the duality between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020148.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020149.png" /> can also be extended to a duality between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020150.png" />-martingales and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020151.png" />-martingales.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020152.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020153.png" />) and a closed subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020154.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020155.png" />) is determined via the following equalities. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020156.png" /> one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020157.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020158.png" />, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020159.png" /> one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020160.png" />, where, as above, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020161.png" /> is two-dimensional Brownian motion starting at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020162.png" />, and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020163.png" />. Then the martingale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020164.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020165.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020166.png" /> is a member of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020167.png" />. The fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020168.png" /> can be considered as a closed subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020169.png" /> is a consequence of the following
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020170.png" /></td> </tr></table>
+
\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*}
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020171.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020172.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020173.png" />.
+
$f \in H _ { 0 } ^ { p }$, $U _ { t } = \operatorname { Re } f ( B _ { t } )$, $U _ { \tau } ^ { * } = \operatorname { sup } _ { 0 \leq t < \tau} | U _ { t } |$.
  
An important equality in the proof of these dualities is the following result: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020174.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020175.png" /> be functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020176.png" />. Then
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020177.png" /></td> </tr></table>
+
\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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020178.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020179.png" />. A similar convention is used for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020180.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020181.png" />. In the first (and in the final) equality, the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020182.png" /> is used: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020183.png" />. The other equalities depend on the fact that a process like <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020184.png" /> is a martingale, which follows from Itô calculus in conjunction with the harmonicity of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020185.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020186.png" />. Next, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020187.png" /> be a function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020188.png" />. Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020189.png" /> the harmonic extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020191.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020192.png" />. Put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020193.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020194.png" /> is a continuous martingale. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020195.png" /> be any stopping time. From the Markov property it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020196.png" />, where
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020197.png" /></td> </tr></table>
+
\begin{equation*} w ( z ) = \int k _ { \vartheta } ( z ) | \varphi ( e ^ { i \vartheta } ) - h ( z ) | ^ { 2 } \frac { d \vartheta } { 2 \pi }, \end{equation*}
  
 
with
 
with
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020198.png" /></td> </tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020199.png" /> can be viewed as the probability density of a Brownian motion starting at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020200.png" /> and exiting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020201.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020202.png" />. Since the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020203.png" /> is equivalent to the inequality
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020204.png" /></td> </tr></table>
+
\begin{equation*} \int _ { I } | \varphi - \varphi _ { I } | ^ { 2 } \frac { d \vartheta } { 2 \pi } \leq c _ { 1 } ^ { 2 } | I |, \end{equation*}
  
for some constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020205.png" />, it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020206.png" /> can be considered as a closed subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020207.png" />: see [[#References|[a6]]], Corol. 2.4; p. 234.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020208.png" /> be function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020209.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020210.png" /> is some interval). Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020211.png" />. Then there exists a pairwise disjoint sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020212.png" /> of open subintervals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020213.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020214.png" /> almost everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020215.png" />,
+
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 $I \backslash \cup I_{j}$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020216.png" /></td> </tr></table>
+
\begin{equation*} \alpha \leq \frac { 1 } { | I _ { j } | } \int _ { I _ { j } } | u ( \vartheta ) | d \vartheta < 2 \alpha, \end{equation*}
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020217.png" /></td> </tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020218.png" /> (see [[#References|[a6]]]), in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020219.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020220.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020221.png" /> is called a Carleson measure if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020222.png" /> for some constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020223.png" /> and for all circle sectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020224.png" />. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020225.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020226.png" /> if and only if
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020227.png" /></td> </tr></table>
+
\begin{equation*} | \nabla u ( z ) | ^ { 2 } \operatorname { log } \frac { 1 } { | z | } d x d y \end{equation*}
  
is a Carleson measure. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020228.png" /> is the harmonic extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020229.png" />. For some other phenomena and related inequalities, see e.g. [[#References|[a3]]], [[#References|[a10]]], and [[#References|[a11]]].
+
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====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020230.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020231.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020232.png" /> dans <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020233.png" /> ou de <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020234.png" /> dans <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020235.png" />"  ''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)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J.B. Garnett,  "Two constructions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020236.png" />"  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)</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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020237.png" /> by singular integral transformations on martingales and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020238.png" />"  ''Math. Scand.'' , '''41'''  (1977)  pp. 140–152</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  P. Koosis,  "Introduction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020239.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020240.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020241.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020242.png" />"  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>
+
<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>

Latest revision as of 05:17, 15 February 2024

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) 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) of a Brownian motion starting at $z \in D$ and exiting $D$ at $e ^ { i \vartheta }$. It is the Poisson kernel (cf. also 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 $\operatorname{BMO}$-space). Here, $\varphi _ { I } = \int _ { I } \varphi d \vartheta / | I |$ and $| I |$ denotes the 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), 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 [a4], p. 200, or [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 [a6], Chap. VI. Composition with the 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 [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). Notice that $B _ { t }$, $t \geq 0$, is a continuous 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

$$ \mathcal{M}^p = \left\{ X : \begin{array}{c} X \text{ a local martingale with respect to } \mathcal{F}, \\ \mathsf{E} |X^*|^P < \infty \end{array} \right\}. $$

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. 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 $X$ 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 [a8]. A martingale $A \in {\cal{M}} ^ { 1 }$ is called an atom if there exists a 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. [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 [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 [a1], [a6], [a7] and [a10], extensions of the above can be found. In particular, some of the concepts can be extended to other domains in $\mathbf{C}$ (see [a6]), in $\mathbf{R} ^ { d }$ and in more general Riemannian manifolds ([a1], [a2], [a7], [a10]). For a relationship with Carleson measures, see [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. [a3], [a10], and [a11].

References

[a1] M. Biroli, U. Mosco, "Sobolev inequalities on homogeneous spaces: Potential theory and degenerate partial differential operators (Parma)" Potential Anal. , 4 (1995) pp. 311–324
[a2] 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
[a3] 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
[a4] R. Durrett, "Brownian motion and martingales in analysis" , Wadsworth (1984) (Contains Mathematica analysis and stochastic processes) MR0750829 Zbl 0554.60075
[a5] 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
[a6] J. Garnett, "Bounded analytic functions" , Acad. Press (1981) MR0628971 Zbl 0469.30024
[a7] R. Hurri-Syrjanen, "The John–Nirenberg inequality and a Sobolev inequality in general domains" J. Math. Anal. Appl. , 175 : 2 (1993) pp. 579–587
[a8] S. Janson, "Characterization of $H ^ { 1 }$ by singular integral transformations on martingales and ${\bf R} ^ { n }$" Math. Scand. , 41 (1977) pp. 140–152
[a9] 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)
[a10] 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
[a11] 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
[a12] 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
How to Cite This Entry:
John-Nirenberg inequalities. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=John-Nirenberg_inequalities&oldid=22609
This article was adapted from an original article by Jan van Casteren (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article