View source for John-Nirenberg inequalities

==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

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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

<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>

Here, the function

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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

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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]]

<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>

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.

==Martingales in Hardy spaces and in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j12002043.png" />.==
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

<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>

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:

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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

<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>

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

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

ii)

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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

<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>

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

<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>

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

<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>

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:

<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>

<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>

<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>

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.

===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

<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>

is valid for all intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j120/j120020/j120020107.png" />.

===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

<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>

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" />.

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.

==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" />.==
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" />.

===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).===
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

<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>

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" />).

===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).===
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" />.

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.

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

<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>

<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" />.

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

<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>

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

<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>

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>

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

<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>

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.

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" />,

<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>

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>

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

<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>

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]]].

====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>