Namespaces
Variants
Actions

Difference between revisions of "VMO-space"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
 +
<!--
 +
v1100502.png
 +
$#A+1 = 34 n = 0
 +
$#C+1 = 34 : ~/encyclopedia/old_files/data/V110/V.1100050 \BMI {fnnme VMO}\EMI\AAhspace,
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
''space of functions of vanishing mean oscillation''
 
''space of functions of vanishing mean oscillation''
  
The class of functions of vanishing mean oscillation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v1100502.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v1100503.png" />, is the subclass of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v1100504.png" /> consisting of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v1100505.png" /> with the property that
+
The class of functions of vanishing mean oscillation on $  \mathbf R  ^ {n} $,  
 +
denoted by $  { \mathop{\rm VMO} } ( \mathbf R  ^ {n} ) $,  
 +
is the subclass of $  { \mathop{\rm BMO} } ( \mathbf R  ^ {n} ) $
 +
consisting of the functions $  f $
 +
with 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/v/v110/v110050/v1100506.png" /></td> </tr></table>
+
$$
 +
{\lim\limits } _ {\left | Q \right | \rightarrow 0 } {
 +
\frac{1}{\left | Q \right | }
 +
} \int\limits _ { Q } {\left | {f - f _ {Q} } \right | }  {dt } \rightarrow 0.
 +
$$
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v1100507.png" /> denotes the volume of the ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v1100508.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v1100509.png" /> denotes the mean of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v11005010.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v11005011.png" /> (see [[BMO-space|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v11005012.png" />-space]]). As with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v11005013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v11005014.png" /> can be defined for spaces of homogeneous type.
+
Here, $  | Q | $
 +
denotes the volume of the ball $  Q $
 +
and $  f _ {Q} $
 +
denotes the mean of $  f $
 +
over $  Q $(
 +
see [[BMO-space| $  { \mathop{\rm BMO} } $-
 +
space]]). As with $  { \mathop{\rm BMO} } $,  
 +
$  { \mathop{\rm VMO} } $
 +
can be defined for spaces of homogeneous type.
  
Some properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v11005015.png" /> are as follows (see also [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]]). Bounded, uniformly continuous functions are in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v11005016.png" /> (cf. [[Uniform continuity|Uniform continuity]]), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v11005017.png" /> can be obtained as the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v11005018.png" />-closure of the continuous functions that vanish at infinity. The [[Hilbert transform|Hilbert transform]] of a bounded, uniformly continuous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v11005019.png" /> is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v11005020.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v11005021.png" /> is the dual of the Hardy space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v11005022.png" /> (cf. also [[Hardy spaces|Hardy spaces]]).
+
Some properties of $  { \mathop{\rm VMO} } $
 +
are as follows (see also [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]]). Bounded, uniformly continuous functions are in $  { \mathop{\rm VMO} } $(
 +
cf. [[Uniform continuity|Uniform continuity]]), and $  { \mathop{\rm VMO} } $
 +
can be obtained as the $  { \mathop{\rm VMO} } $-
 +
closure of the continuous functions that vanish at infinity. The [[Hilbert transform|Hilbert transform]] of a bounded, uniformly continuous function on $  \mathbf R $
 +
is in $  { \mathop{\rm VMO} } ( \mathbf R ) $.  
 +
$  { \mathop{\rm VMO} } ( \mathbf R  ^ {n} ) $
 +
is the dual of the Hardy space $  H  ^ {1} ( \mathbf R  ^ {n} ) $(
 +
cf. also [[Hardy spaces|Hardy spaces]]).
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v11005023.png" /> appears in the theory of Douglas algebras: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v11005024.png" /> be the boundary of the unit disc in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v11005025.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v11005026.png" /> denote the subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v11005027.png" /> consisting of the boundary values of bounded holomorphic functions (cf. [[Analytic function|Analytic function]]) on the unit disc and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v11005028.png" /> denote the set of continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v11005029.png" />. Put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v11005030.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v11005031.png" /> is a closed subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v11005032.png" /> and the simplest example of a Douglas algebra. Its largest self-adjoint subalgebra, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v11005033.png" />, equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v11005034.png" />, [[#References|[a2]]], [[#References|[a3]]].
+
$  { \mathop{\rm VMO} } $
 +
appears in the theory of Douglas algebras: Let $  T $
 +
be the boundary of the unit disc in $  \mathbf C $.  
 +
Let $  H  ^  \infty  $
 +
denote the subspace of $  L _  \infty  ( T ) $
 +
consisting of the boundary values of bounded holomorphic functions (cf. [[Analytic function|Analytic function]]) on the unit disc and let $  C $
 +
denote the set of continuous functions on $  T $.  
 +
Put $  H  ^  \infty  + C = \{ {f + g } : {f \in H  ^  \infty  , g \in C } \} $.  
 +
$  H  ^  \infty  + C $
 +
is a closed subalgebra of $  L _  \infty  ( T ) $
 +
and the simplest example of a Douglas algebra. Its largest self-adjoint subalgebra, $  QC $,  
 +
equals $  L _  \infty  ( T ) \cap { \mathop{\rm VMO} } ( T ) $,  
 +
[[#References|[a2]]], [[#References|[a3]]].
  
See also [[BMO-space|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110050/v11005035.png" />-space]].
+
See also [[BMO-space| $  { \mathop{\rm BMO} } $-
 +
space]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</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. 569–645</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Garnett,  "Bounded analytic functions" , Acad. Press  (1981)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Sarason,  "Functions of vanishing mean oscillation"  ''Trans. Amer. Math. Soc.'' , '''207'''  (1975)  pp. 391–405</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</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. 569–645</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Garnett,  "Bounded analytic functions" , Acad. Press  (1981)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Sarason,  "Functions of vanishing mean oscillation"  ''Trans. Amer. Math. Soc.'' , '''207'''  (1975)  pp. 391–405</TD></TR></table>

Revision as of 08:27, 6 June 2020


space of functions of vanishing mean oscillation

The class of functions of vanishing mean oscillation on $ \mathbf R ^ {n} $, denoted by $ { \mathop{\rm VMO} } ( \mathbf R ^ {n} ) $, is the subclass of $ { \mathop{\rm BMO} } ( \mathbf R ^ {n} ) $ consisting of the functions $ f $ with the property that

$$ {\lim\limits } _ {\left | Q \right | \rightarrow 0 } { \frac{1}{\left | Q \right | } } \int\limits _ { Q } {\left | {f - f _ {Q} } \right | } {dt } \rightarrow 0. $$

Here, $ | Q | $ denotes the volume of the ball $ Q $ and $ f _ {Q} $ denotes the mean of $ f $ over $ Q $( see $ { \mathop{\rm BMO} } $- space). As with $ { \mathop{\rm BMO} } $, $ { \mathop{\rm VMO} } $ can be defined for spaces of homogeneous type.

Some properties of $ { \mathop{\rm VMO} } $ are as follows (see also [a1], [a2], [a3]). Bounded, uniformly continuous functions are in $ { \mathop{\rm VMO} } $( cf. Uniform continuity), and $ { \mathop{\rm VMO} } $ can be obtained as the $ { \mathop{\rm VMO} } $- closure of the continuous functions that vanish at infinity. The Hilbert transform of a bounded, uniformly continuous function on $ \mathbf R $ is in $ { \mathop{\rm VMO} } ( \mathbf R ) $. $ { \mathop{\rm VMO} } ( \mathbf R ^ {n} ) $ is the dual of the Hardy space $ H ^ {1} ( \mathbf R ^ {n} ) $( cf. also Hardy spaces).

$ { \mathop{\rm VMO} } $ appears in the theory of Douglas algebras: Let $ T $ be the boundary of the unit disc in $ \mathbf C $. Let $ H ^ \infty $ denote the subspace of $ L _ \infty ( T ) $ consisting of the boundary values of bounded holomorphic functions (cf. Analytic function) on the unit disc and let $ C $ denote the set of continuous functions on $ T $. Put $ H ^ \infty + C = \{ {f + g } : {f \in H ^ \infty , g \in C } \} $. $ H ^ \infty + C $ is a closed subalgebra of $ L _ \infty ( T ) $ and the simplest example of a Douglas algebra. Its largest self-adjoint subalgebra, $ QC $, equals $ L _ \infty ( T ) \cap { \mathop{\rm VMO} } ( T ) $, [a2], [a3].

See also $ { \mathop{\rm BMO} } $- space.

References

[a1] R.R. Coifman, G. Weiss, "Extensions of Hardy-spaces and their use in analysis" Bull. Amer. Math. Soc. , 83 (1977) pp. 569–645
[a2] J. Garnett, "Bounded analytic functions" , Acad. Press (1981)
[a3] D. Sarason, "Functions of vanishing mean oscillation" Trans. Amer. Math. Soc. , 207 (1975) pp. 391–405
How to Cite This Entry:
VMO-space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=VMO-space&oldid=18125
This article was adapted from an original article by J. Wiegerinck (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article