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''space of functions of bounded mean oscillation''
 
''space of functions of bounded mean oscillation''
  
Functions of bounded mean oscillation were introduced by F. John and L. Nirenberg [[#References|[a8]]], [[#References|[a12]]], in connection with differential equations. The definition on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b1106602.png" /> reads as follows: Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b1106603.png" /> is integrable over compact sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b1106604.png" />, (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b1106605.png" />), and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b1106606.png" /> is any ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b1106607.png" />, with volume denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b1106608.png" />. The mean of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b1106609.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066010.png" /> will be
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Functions of bounded mean oscillation were introduced by F. John and L. Nirenberg [[#References|[a8]]], [[#References|[a12]]], in connection with differential equations. The definition on ${\bf R} ^ { n }$ reads as follows: Suppose that $f$ is integrable over compact sets in ${\bf R} ^ { n }$, (i.e. $f \in L ^ { 1 _ {\operatorname{ loc }}} ( \mathbf{R} )$), and that $Q$ is any ball in ${\bf R} ^ { n }$, with volume denoted by $|Q|$. The mean of $f$ over $Q$ will be
  
<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/b/b110/b110660/b11066011.png" /></td> </tr></table>
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\begin{equation*} f _ { Q } = \frac { 1 } { | Q | } \int _ { Q } f ( t ) d t. \end{equation*}
  
By definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066012.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066013.png" /> if
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By definition, $f$ belongs to $\operatorname{BMO}$ 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/b/b110/b110660/b11066014.png" /></td> </tr></table>
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\begin{equation*} \| f \| _ { * } = \operatorname { sup } _ { Q } \frac { 1 } { | Q | } \int _ { Q } | f ( t ) - f _ { Q } | d t &lt; \infty, \end{equation*}
  
where the supremum is taken over all balls <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066015.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066016.png" /> is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066018.png" />-norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066019.png" />, and it becomes a [[Norm|norm]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066020.png" /> after dividing out the constant functions. Bounded functions are in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066021.png" /> and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066022.png" />-function is locally in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066023.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066024.png" />. Typical examples of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066025.png" />-functions are of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066026.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066027.png" /> a [[Polynomial|polynomial]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066028.png" />.
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where the supremum is taken over all balls $Q$. Here, $\| f \|_*$ is called the $\operatorname{BMO}$-norm of $f$, and it becomes a [[Norm|norm]] on $\operatorname{BMO}$ after dividing out the constant functions. Bounded functions are in $\operatorname{BMO}$ and a $\operatorname{BMO}$-function is locally in $L _ { p } ( \mathbf{R} )$ for every $p &lt; \infty$. Typical examples of $\operatorname{BMO}$-functions are of the form $\operatorname { log } | P |$ with $P$ a [[Polynomial|polynomial]] on ${\bf R} ^ { n }$.
  
The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066029.png" /> is very important in modern [[Harmonic analysis|harmonic analysis]]. Taking <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066030.png" />, the [[Hilbert transform|Hilbert transform]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066031.png" />, defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066032.png" />, maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066033.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066034.png" /> boundedly, i.e.
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The space $\operatorname{BMO}$ is very important in modern [[Harmonic analysis|harmonic analysis]]. Taking $n = 1$, the [[Hilbert transform|Hilbert transform]] $H$, defined by $H f ( x ) = \operatorname { lim } _ { \epsilon \downarrow 0} \int _ { | t | &gt; \epsilon } f ( x - t ) / t d t$, maps $L _ { \infty } ( \mathbf{R} )$ to $\operatorname{BMO}$ boundedly, i.e.
  
<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/b/b110/b110660/b11066035.png" /></td> </tr></table>
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\begin{equation*} \| H f \| _ { * } \leq G \| f \| _ { \infty }. \end{equation*}
  
The same is true for a large class of singular integral transformations (cf. also [[Singular integral|Singular integral]]), including Riesz transformations [[#References|[a12]]]. There is a version of the Riesz interpolation theorem (cf. also [[Riesz interpolation formula|Riesz interpolation formula]]) for analytic families of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066037.png" />, which besides the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066038.png" />-boundedness assumptions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066039.png" /> involves the (weak) assumption <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066040.png" /> instead of the usual assumption <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066041.png" />, cf. [[#References|[a12]]]. However the most famous result is the Fefferman duality theorem, [[#References|[a6]]], [[#References|[a7]]], [[#References|[a12]]]. It states that the dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066042.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066043.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066044.png" /> denotes the real Hardy space on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066045.png" /> (cf. also [[Hardy spaces|Hardy spaces]]). The result is also valid for the usual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066046.png" /> on the disc or the upper half-plane, with an appropriate complex multiplication on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066047.png" />, cf. [[#References|[a5]]].
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The same is true for a large class of singular integral transformations (cf. also [[Singular integral|Singular integral]]), including Riesz transformations [[#References|[a12]]]. There is a version of the Riesz interpolation theorem (cf. also [[Riesz interpolation formula|Riesz interpolation formula]]) for analytic families of operators $\{ T _ { s } \}$, $0 \leq \operatorname { Re } s \leq 1$, which besides the $L_{2}$-boundedness assumptions on $\| T _ { i t } \|$ involves the (weak) assumption $\| T _ { 1 } + i t ( f ) \| _ { * } \leq C \| f \| _ { \infty }$ instead of the usual assumption $\| T _ { 1  + i t} ( f ) \| _ { \infty } \leq C \| f \|_\infty$, cf. [[#References|[a12]]]. However the most famous result is the Fefferman duality theorem, [[#References|[a6]]], [[#References|[a7]]], [[#References|[a12]]]. It states that the dual of $H ^ { 1 }$ is $\operatorname{BMO}$. Here, $H ^ { 1 }$ denotes the real Hardy space on ${\bf R} ^ { n }$ (cf. also [[Hardy spaces|Hardy spaces]]). The result is also valid for the usual space $H ^ { 1 }$ on the disc or the upper half-plane, with an appropriate complex multiplication on $\operatorname{BMO}$, cf. [[#References|[a5]]].
  
Calderón–Zygmund operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066048.png" /> form an important class of singular integral operators. A Calderón–Zygmund operator can be defined as a [[Linear operator|linear operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066049.png" /> with associated [[Schwarz kernel|Schwarz kernel]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066050.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066051.png" /> with the following properties:
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Calderón–Zygmund operators on ${\bf R} ^ { n }$ form an important class of singular integral operators. A Calderón–Zygmund operator can be defined as a [[Linear operator|linear operator]] $T : \mathcal{D} ( \mathbf{R} ^ { n } ) \rightarrow \mathcal{D}  ^ { \prime } ( \mathbf{R} ^ { n } )$ with associated [[Schwarz kernel|Schwarz kernel]] $K ( x , y )$ defined on $\Omega = \{ ( x , y ) : x , y \in \mathbf{R} ^ { n } , x \neq y \}$ with the following properties:
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066052.png" /> is locally integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066053.png" /> and satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066054.png" />;
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i) $K$ is locally integrable on $\Omega$ and satisfies $| K ( x , y ) | = O ( | x - y | ^ { - x } )$;
  
ii) there exist constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066056.png" /> such that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066058.png" />,
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ii) there exist constants $C &gt; 0$ and $0 &lt; \gamma \leq 1$ such that for $( x , y ) \in \Omega$ and $| x ^ { \prime } - x | \leq | x - y | / 2$,
  
<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/b/b110/b110660/b11066059.png" /></td> </tr></table>
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\begin{equation*} | K ( x - , y ) - K ( x , y ) | \leq C | x ^ { \prime } - x | ^ { \gamma } | x - y | ^ { - n - \gamma }. \end{equation*}
  
Similarly, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066061.png" />,
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Similarly, for $( x , y ) \in \Omega$ and $| y ^ { \prime } - y | \leq | x - y | / 2$,
  
<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/b/b110/b110660/b11066062.png" /></td> </tr></table>
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\begin{equation*} | K ( x , y ^ { \prime } ) - K ( x , y ) | \leq C | y ^ { \prime } - y | ^ { \gamma } | x - y | ^ { - n - \gamma }. \end{equation*}
  
iii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066063.png" /> can be extended to a bounded linear operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066064.png" />.
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iii) $T$ can be extended to a bounded linear operator on $L _ { 2 } ( \mathbf{R} ^ { n } )$.
  
This last condition is hard to verify in general. Thus, it is an important result, known as the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066066.png" />-theorem, that if i) and ii) hold, then iii) is equivalent to: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066067.png" /> is weakly bounded on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066068.png" /> and both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066069.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066070.png" /> are in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066071.png" />, cf. [[#References|[a3]]], [[#References|[a11]]], [[#References|[a12]]]. It is known that diagonal operators with respect to an orthonormal wavelet basis are of Calderón–Zygmund type. This connection with [[Wavelet analysis|wavelet analysis]] is treated in [[#References|[a11]]].
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This last condition is hard to verify in general. Thus, it is an important result, known as the $T ( 1 )$-theorem, that if i) and ii) hold, then iii) is equivalent to: $T$ is weakly bounded on $L_{2}$ and both $T ( 1 )$ and $T ^ { * } ( 1 )$ are in $\operatorname{BMO}$, cf. [[#References|[a3]]], [[#References|[a11]]], [[#References|[a12]]]. It is known that diagonal operators with respect to an orthonormal wavelet basis are of Calderón–Zygmund type. This connection with [[Wavelet analysis|wavelet analysis]] is treated in [[#References|[a11]]].
  
Many of the results concerning <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066072.png" />-functions have been generalized to the setting of martingales, cf. [[#References|[a9]]] (see also [[Martingale|Martingale]]).
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Many of the results concerning $\operatorname{BMO}$-functions have been generalized to the setting of martingales, cf. [[#References|[a9]]] (see also [[Martingale|Martingale]]).
  
The duality result indicates that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066073.png" /> plays a role in complex analysis as well. The class of holomorphic functions (cf. [[Analytic function|Analytic function]]) on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066074.png" /> with boundary values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066075.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066077.png" />, and is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066079.png" />-space, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066080.png" />.
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The duality result indicates that $\operatorname{BMO}$ plays a role in complex analysis as well. The class of holomorphic functions (cf. [[Analytic function|Analytic function]]) on a domain $D$ with boundary values in $\operatorname{BMO}$ is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066077.png"/>, and is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066079.png"/>-space, i.e., $= \operatorname{BMOA}= \operatorname{BMO} \cap H ^ { 2 }$.
  
Carleson's corona theorem [[#References|[a5]]] for the disc states that for given bounded holomorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066081.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066082.png" /> there exist bounded holomorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066083.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066084.png" />. So far (1996), this result could not be extended to the unit ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066086.png" />, but it can be proved if one only requires that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066087.png" />, cf. [[#References|[a13]]].
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Carleson's corona theorem [[#References|[a5]]] for the disc states that for given bounded holomorphic functions $f _ { 1 } , \ldots , f _ { n }$ such that $\sum _ { i } | f _ { i } | &gt; \delta &gt; 0$ there exist bounded holomorphic functions $g_ 1 , \ldots , g_ { n }$ such that $\sum _ { i } f _ { i } g _ { i } = 1$. So far (1996), this result could not be extended to the unit ball in $\mathbf{C}^{m}$, $m &gt; 1$, but it can be proved if one only requires that $g_i \in \operatorname { BMOA}$, cf. [[#References|[a13]]].
  
The definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066088.png" /> makes sense as soon as there are proper notions of integral and ball in a space. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066089.png" /> can be defined in spaces of homogeneous type, cf. [[#References|[a1]]], [[#References|[a2]]], [[#References|[a10]]]. In the setting of several complex variables, several types of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066090.png" />-spaces arise on the boundary of (strictly) pseudoconvex domains, depending on whether one considers the isotropic Euclidean balls or the non-isotropic balls that are natural in connection with pseudo-convexity, cf. [[#References|[a10]]].
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The definition of $\operatorname{BMO}$ makes sense as soon as there are proper notions of integral and ball in a space. Thus, $\operatorname{BMO}$ can be defined in spaces of homogeneous type, cf. [[#References|[a1]]], [[#References|[a2]]], [[#References|[a10]]]. In the setting of several complex variables, several types of $\operatorname{BMO}$-spaces arise on the boundary of (strictly) pseudoconvex domains, depending on whether one considers the isotropic Euclidean balls or the non-isotropic balls that are natural in connection with pseudo-convexity, cf. [[#References|[a10]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.R. Coifman,  G. Weiss,  "Analyse harmonique non-commutative sur certains espaces homogènes" , ''Lecture Notes in Mathematics'' , '''242''' , Springer  (1971)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.R. Coifman,  G. Weiss,  "Extensions of Hardy spaces and their use in analysis"  ''Bull. Amer. Math. Soc.'' , '''83'''  (1977)  pp. 569–643</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. David,  J.-L. Journé,  "A boundedness criterion for generalized Calderón–Zygmund operators"  ''Ann. of Math.'' , '''120'''  (1985)  pp. 371–397</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Garcia-Cuervas,  J.L. Rubio de Francia,  "Weighted norm inequalities and related topics" , ''Math. Stud.'' , '''116''' , North-Holland  (1985)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J. Garnett,  "Bounded analytic functions" , Acad. Press  (1981)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  C. Fefferman,  "Characterizations of bounded mean oscillation"  ''Bull. Amer. Math. Soc.'' , '''77'''  (1971)  pp. 587–588</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  C. Fefferman,  E.M. Stein,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066091.png" /> spaces of several variables"  ''Acta Math.'' , '''129'''  (1974)  pp. 137–193</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  F. John,  L. Nirenberg,  "On functions of bounded mean oscillation"  ''Comm. Pure Appl. Math.'' , '''14'''  (1961)  pp. 415–426</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  N. Kazamaki,  "Continuous exponential martingales and BMO" , ''Lecture Notes in Mathematics'' , '''579''' , Springer  (1994)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  S.G. Krantz,  "Geometric analysis and function spaces" , ''CBMS'' , '''81''' , Amer. Math. Soc.  (1993)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  Y. Meyer,  "Ondelettes et opérateurs II. Opérateurs de Calderón–Zygmund" , ''Actual. Math.'' , Hermann  (1990)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  E.M. Stein,  "Harmonic analysis: real variable methods, orthogonality, and oscillatory integrals" , ''Math. Ser.'' , '''43''' , Princeton Univ. Press  (1993)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  N.Th. Varopoulos,  "BMO functions and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110660/b11066092.png" /> equation"  ''Pacific J. Math.'' , '''71'''  (1977)  pp. 221–272</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  R.R. Coifman,  G. Weiss,  "Analyse harmonique non-commutative sur certains espaces homogènes" , ''Lecture Notes in Mathematics'' , '''242''' , Springer  (1971)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  R.R. Coifman,  G. Weiss,  "Extensions of Hardy spaces and their use in analysis"  ''Bull. Amer. Math. Soc.'' , '''83'''  (1977)  pp. 569–643</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  G. David,  J.-L. Journé,  "A boundedness criterion for generalized Calderón–Zygmund operators"  ''Ann. of Math.'' , '''120'''  (1985)  pp. 371–397</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  J. Garcia-Cuervas,  J.L. Rubio de Francia,  "Weighted norm inequalities and related topics" , ''Math. Stud.'' , '''116''' , North-Holland  (1985)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  J. Garnett,  "Bounded analytic functions" , Acad. Press  (1981)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  C. Fefferman,  "Characterizations of bounded mean oscillation"  ''Bull. Amer. Math. Soc.'' , '''77'''  (1971)  pp. 587–588</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  C. Fefferman,  E.M. Stein,  "$H ^ { p }$ spaces of several variables"  ''Acta Math.'' , '''129'''  (1974)  pp. 137–193</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  F. John,  L. Nirenberg,  "On functions of bounded mean oscillation"  ''Comm. Pure Appl. Math.'' , '''14'''  (1961)  pp. 415–426</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  N. Kazamaki,  "Continuous exponential martingales and BMO" , ''Lecture Notes in Mathematics'' , '''579''' , Springer  (1994)</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  S.G. Krantz,  "Geometric analysis and function spaces" , ''CBMS'' , '''81''' , Amer. Math. Soc.  (1993)</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  Y. Meyer,  "Ondelettes et opérateurs II. Opérateurs de Calderón–Zygmund" , ''Actual. Math.'' , Hermann  (1990)</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  E.M. Stein,  "Harmonic analysis: real variable methods, orthogonality, and oscillatory integrals" , ''Math. Ser.'' , '''43''' , Princeton Univ. Press  (1993)</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  N.Th. Varopoulos,  "BMO functions and the $\partial$ equation"  ''Pacific J. Math.'' , '''71'''  (1977)  pp. 221–272</td></tr></table>

Revision as of 16:57, 1 July 2020

space of functions of bounded mean oscillation

Functions of bounded mean oscillation were introduced by F. John and L. Nirenberg [a8], [a12], in connection with differential equations. The definition on ${\bf R} ^ { n }$ reads as follows: Suppose that $f$ is integrable over compact sets in ${\bf R} ^ { n }$, (i.e. $f \in L ^ { 1 _ {\operatorname{ loc }}} ( \mathbf{R} )$), and that $Q$ is any ball in ${\bf R} ^ { n }$, with volume denoted by $|Q|$. The mean of $f$ over $Q$ will be

\begin{equation*} f _ { Q } = \frac { 1 } { | Q | } \int _ { Q } f ( t ) d t. \end{equation*}

By definition, $f$ belongs to $\operatorname{BMO}$ if

\begin{equation*} \| f \| _ { * } = \operatorname { sup } _ { Q } \frac { 1 } { | Q | } \int _ { Q } | f ( t ) - f _ { Q } | d t < \infty, \end{equation*}

where the supremum is taken over all balls $Q$. Here, $\| f \|_*$ is called the $\operatorname{BMO}$-norm of $f$, and it becomes a norm on $\operatorname{BMO}$ after dividing out the constant functions. Bounded functions are in $\operatorname{BMO}$ and a $\operatorname{BMO}$-function is locally in $L _ { p } ( \mathbf{R} )$ for every $p < \infty$. Typical examples of $\operatorname{BMO}$-functions are of the form $\operatorname { log } | P |$ with $P$ a polynomial on ${\bf R} ^ { n }$.

The space $\operatorname{BMO}$ is very important in modern harmonic analysis. Taking $n = 1$, the Hilbert transform $H$, defined by $H f ( x ) = \operatorname { lim } _ { \epsilon \downarrow 0} \int _ { | t | > \epsilon } f ( x - t ) / t d t$, maps $L _ { \infty } ( \mathbf{R} )$ to $\operatorname{BMO}$ boundedly, i.e.

\begin{equation*} \| H f \| _ { * } \leq G \| f \| _ { \infty }. \end{equation*}

The same is true for a large class of singular integral transformations (cf. also Singular integral), including Riesz transformations [a12]. There is a version of the Riesz interpolation theorem (cf. also Riesz interpolation formula) for analytic families of operators $\{ T _ { s } \}$, $0 \leq \operatorname { Re } s \leq 1$, which besides the $L_{2}$-boundedness assumptions on $\| T _ { i t } \|$ involves the (weak) assumption $\| T _ { 1 } + i t ( f ) \| _ { * } \leq C \| f \| _ { \infty }$ instead of the usual assumption $\| T _ { 1 + i t} ( f ) \| _ { \infty } \leq C \| f \|_\infty$, cf. [a12]. However the most famous result is the Fefferman duality theorem, [a6], [a7], [a12]. It states that the dual of $H ^ { 1 }$ is $\operatorname{BMO}$. Here, $H ^ { 1 }$ denotes the real Hardy space on ${\bf R} ^ { n }$ (cf. also Hardy spaces). The result is also valid for the usual space $H ^ { 1 }$ on the disc or the upper half-plane, with an appropriate complex multiplication on $\operatorname{BMO}$, cf. [a5].

Calderón–Zygmund operators on ${\bf R} ^ { n }$ form an important class of singular integral operators. A Calderón–Zygmund operator can be defined as a linear operator $T : \mathcal{D} ( \mathbf{R} ^ { n } ) \rightarrow \mathcal{D} ^ { \prime } ( \mathbf{R} ^ { n } )$ with associated Schwarz kernel $K ( x , y )$ defined on $\Omega = \{ ( x , y ) : x , y \in \mathbf{R} ^ { n } , x \neq y \}$ with the following properties:

i) $K$ is locally integrable on $\Omega$ and satisfies $| K ( x , y ) | = O ( | x - y | ^ { - x } )$;

ii) there exist constants $C > 0$ and $0 < \gamma \leq 1$ such that for $( x , y ) \in \Omega$ and $| x ^ { \prime } - x | \leq | x - y | / 2$,

\begin{equation*} | K ( x - , y ) - K ( x , y ) | \leq C | x ^ { \prime } - x | ^ { \gamma } | x - y | ^ { - n - \gamma }. \end{equation*}

Similarly, for $( x , y ) \in \Omega$ and $| y ^ { \prime } - y | \leq | x - y | / 2$,

\begin{equation*} | K ( x , y ^ { \prime } ) - K ( x , y ) | \leq C | y ^ { \prime } - y | ^ { \gamma } | x - y | ^ { - n - \gamma }. \end{equation*}

iii) $T$ can be extended to a bounded linear operator on $L _ { 2 } ( \mathbf{R} ^ { n } )$.

This last condition is hard to verify in general. Thus, it is an important result, known as the $T ( 1 )$-theorem, that if i) and ii) hold, then iii) is equivalent to: $T$ is weakly bounded on $L_{2}$ and both $T ( 1 )$ and $T ^ { * } ( 1 )$ are in $\operatorname{BMO}$, cf. [a3], [a11], [a12]. It is known that diagonal operators with respect to an orthonormal wavelet basis are of Calderón–Zygmund type. This connection with wavelet analysis is treated in [a11].

Many of the results concerning $\operatorname{BMO}$-functions have been generalized to the setting of martingales, cf. [a9] (see also Martingale).

The duality result indicates that $\operatorname{BMO}$ plays a role in complex analysis as well. The class of holomorphic functions (cf. Analytic function) on a domain $D$ with boundary values in $\operatorname{BMO}$ is denoted by , and is called the -space, i.e., $= \operatorname{BMOA}= \operatorname{BMO} \cap H ^ { 2 }$.

Carleson's corona theorem [a5] for the disc states that for given bounded holomorphic functions $f _ { 1 } , \ldots , f _ { n }$ such that $\sum _ { i } | f _ { i } | > \delta > 0$ there exist bounded holomorphic functions $g_ 1 , \ldots , g_ { n }$ such that $\sum _ { i } f _ { i } g _ { i } = 1$. So far (1996), this result could not be extended to the unit ball in $\mathbf{C}^{m}$, $m > 1$, but it can be proved if one only requires that $g_i \in \operatorname { BMOA}$, cf. [a13].

The definition of $\operatorname{BMO}$ makes sense as soon as there are proper notions of integral and ball in a space. Thus, $\operatorname{BMO}$ can be defined in spaces of homogeneous type, cf. [a1], [a2], [a10]. In the setting of several complex variables, several types of $\operatorname{BMO}$-spaces arise on the boundary of (strictly) pseudoconvex domains, depending on whether one considers the isotropic Euclidean balls or the non-isotropic balls that are natural in connection with pseudo-convexity, cf. [a10].

References

[a1] R.R. Coifman, G. Weiss, "Analyse harmonique non-commutative sur certains espaces homogènes" , Lecture Notes in Mathematics , 242 , Springer (1971)
[a2] R.R. Coifman, G. Weiss, "Extensions of Hardy spaces and their use in analysis" Bull. Amer. Math. Soc. , 83 (1977) pp. 569–643
[a3] G. David, J.-L. Journé, "A boundedness criterion for generalized Calderón–Zygmund operators" Ann. of Math. , 120 (1985) pp. 371–397
[a4] J. Garcia-Cuervas, J.L. Rubio de Francia, "Weighted norm inequalities and related topics" , Math. Stud. , 116 , North-Holland (1985)
[a5] J. Garnett, "Bounded analytic functions" , Acad. Press (1981)
[a6] C. Fefferman, "Characterizations of bounded mean oscillation" Bull. Amer. Math. Soc. , 77 (1971) pp. 587–588
[a7] C. Fefferman, E.M. Stein, "$H ^ { p }$ spaces of several variables" Acta Math. , 129 (1974) pp. 137–193
[a8] F. John, L. Nirenberg, "On functions of bounded mean oscillation" Comm. Pure Appl. Math. , 14 (1961) pp. 415–426
[a9] N. Kazamaki, "Continuous exponential martingales and BMO" , Lecture Notes in Mathematics , 579 , Springer (1994)
[a10] S.G. Krantz, "Geometric analysis and function spaces" , CBMS , 81 , Amer. Math. Soc. (1993)
[a11] Y. Meyer, "Ondelettes et opérateurs II. Opérateurs de Calderón–Zygmund" , Actual. Math. , Hermann (1990)
[a12] E.M. Stein, "Harmonic analysis: real variable methods, orthogonality, and oscillatory integrals" , Math. Ser. , 43 , Princeton Univ. Press (1993)
[a13] N.Th. Varopoulos, "BMO functions and the $\partial$ equation" Pacific J. Math. , 71 (1977) pp. 221–272
How to Cite This Entry:
BMO-space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=BMO-space&oldid=19103
This article was adapted from an original article by J. Wiegerinck (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article