Namespaces
Variants
Actions

Difference between revisions of "BMOA-space"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (AUTOMATIC EDIT (latexlist): Replaced 14 formulas out of 24 by TEX code with an average confidence of 2.0 and a minimal confidence of 2.0.)
Line 1: Line 1:
 +
<!--This article has been texified automatically. Since there was no Nroff source code for this article,
 +
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
 +
was used.
 +
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
 +
 +
Out of 24 formulas, 14 were replaced by TEX code.-->
 +
 +
{{TEX|semi-auto}}{{TEX|partial}}
 
''space of analytic functions of bounded mean oscillation''
 
''space of analytic functions of bounded mean oscillation''
  
In 1961, F. John and L. Nirenberg [[#References|[a4]]] introduced the space of functions of bounded mean oscillation, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b1301802.png" />, in their study of differential equations (cf. also [[BMO-space|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b1301803.png" />-space]]). About a decade later, C. Fefferman proved his famous duality theorem [[#References|[a1]]] [[#References|[a2]]], which states that the dual of the Hardy space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b1301804.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b1301805.png" /> (cf. also [[Hardy spaces|Hardy spaces]]).
+
In 1961, F. John and L. Nirenberg [[#References|[a4]]] introduced the space of functions of bounded mean oscillation, $\operatorname{BMO}$, in their study of differential equations (cf. also [[BMO-space|$\operatorname{BMO}$-space]]). About a decade later, C. Fefferman proved his famous duality theorem [[#References|[a1]]] [[#References|[a2]]], which states that the dual of the Hardy space $H ^ { 1 }$ is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b1301805.png"/> (cf. also [[Hardy spaces|Hardy spaces]]).
  
In these early works, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b1301806.png" /> was studied primarily as a space of real-valued functions, but Fefferman's result raised questions about the nature of analytic functions in the Hardy spaces of the unit disc whose boundary values are in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b1301807.png" />. This is the definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b1301808.png" /> and the duality theorem provides the alternative that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b1301809.png" /> consists of those analytic functions that can be represented as a sum of two analytic functions, one with a bounded real part and the other with a bounded imaginary part (cf. also [[Analytic function|Analytic function]]; [[Hardy spaces|Hardy spaces]]).
+
In these early works, $\operatorname{BMO}$ was studied primarily as a space of real-valued functions, but Fefferman's result raised questions about the nature of analytic functions in the Hardy spaces of the unit disc whose boundary values are in $\operatorname{BMO}$. This is the definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b1301808.png"/> and the duality theorem provides the alternative that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b1301809.png"/> consists of those analytic functions that can be represented as a sum of two analytic functions, one with a bounded real part and the other with a bounded imaginary part (cf. also [[Analytic function|Analytic function]]; [[Hardy spaces|Hardy spaces]]).
  
Ch. Pommerenke [[#References|[a5]]] proved that a [[Univalent function|univalent function]] is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b13018010.png" /> if and only if there is a bound on the radius of the discs contained in the image. Subsequently, W. Hayman and Pommerenke [[#References|[a3]]] and D. Stegenga [[#References|[a7]]] proved that any analytic function is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b13018011.png" /> provided the complement of its image is sufficiently thick in a technical sense that uses the notion of [[Logarithmic capacity|logarithmic capacity]]. As an example, any function whose image does not contain a disc of a fixed radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b13018012.png" /> centred at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b13018013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b13018014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b13018015.png" /> range over all integers, satisfies this criterion and hence is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b13018016.png" />.
+
Ch. Pommerenke [[#References|[a5]]] proved that a [[Univalent function|univalent function]] is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b13018010.png"/> if and only if there is a bound on the radius of the discs contained in the image. Subsequently, W. Hayman and Pommerenke [[#References|[a3]]] and D. Stegenga [[#References|[a7]]] proved that any analytic function is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b13018011.png"/> provided the complement of its image is sufficiently thick in a technical sense that uses the notion of [[Logarithmic capacity|logarithmic capacity]]. As an example, any function whose image does not contain a disc of a fixed radius $\epsilon &gt; 0$ centred at $a + i b$, where $a$, $b$ range over all integers, satisfies this criterion and hence is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b13018016.png"/>.
  
In a similar vein, K. Stephenson and Stegenga [[#References|[a6]]] proved that an analytic function is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b13018017.png" /> provided its image [[Riemann surface|Riemann surface]] (viewed as spread out over the complex plane) has the following property: There are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b13018018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b13018019.png" /> so that a Brownian traveller will, with probability at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b13018020.png" />, fall off the edge of the surface before travelling outward <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b13018021.png" /> units (cf. also [[Brownian motion|Brownian motion]]). As an example, an overlapping infinite saussage-shaped region can be constructed so that the Riemann mapping function maps onto to the entire complex plane but is nevertheless in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b13018022.png" />.
+
In a similar vein, K. Stephenson and Stegenga [[#References|[a6]]] proved that an analytic function is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b13018017.png"/> provided its image [[Riemann surface|Riemann surface]] (viewed as spread out over the complex plane) has the following property: There are $0 &lt; R &lt; \infty$, $0 &lt; \epsilon &lt; 1$ so that a Brownian traveller will, with probability at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b13018020.png"/>, fall off the edge of the surface before travelling outward $R$ units (cf. also [[Brownian motion|Brownian motion]]). As an example, an overlapping infinite saussage-shaped region can be constructed so that the Riemann mapping function maps onto to the entire complex plane but is nevertheless in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b13018022.png"/>.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b13018023.png" /> comes up naturally in many problems in analysis, such as on the composition operator, the corona problem (cf. also [[Hardy classes|Hardy classes]]), and on functions in one and several complex variables. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b13018024.png" /> and its variants has become an indispensable tool in real and complex analysis.
+
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b13018023.png"/> comes up naturally in many problems in analysis, such as on the composition operator, the corona problem (cf. also [[Hardy classes|Hardy classes]]), and on functions in one and several complex variables. $\operatorname{BMO}$ and its variants has become an indispensable tool in real and complex analysis.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Fefferman,  "Characterization of bounded mean oscillation"  ''Bull. Amer. Math. Soc.'' , '''77'''  (1971)  pp. 587–588</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Fefferman,  E. Stein,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130180/b13018025.png" /> spaces of several variables"  ''Acta Math.'' , '''129'''  (1974)  pp. 137–193</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Hayman,  Ch. Pommerenke,  "On analytic functions of bounded mean oscillation"  ''Bull. London Math. Soc.'' , '''10'''  (1978)  pp. 219–224</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  F. John,  L. Nirenberg,  "On functions of bounded mean oscillation"  ''Commun. Pure Appl. Math.'' , '''14'''  (1961)  pp. 415–426</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  Ch. Pommerenke,  "Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation"  ''Comment. Math. Helv.'' , '''152'''  (1977)  pp. 591–602</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  K. Stephenson,  D. Stegenga,  "A geometric characterization of analytic functions of bounded mean oscillation"  ''J. London Math. Soc. (2)'' , '''24'''  (1981)  pp. 243–254</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  D. Stegenga,  "A geometric condition that implies BMOA" , ''Proc. Symp. Pure Math.'' , '''XXXV:1''' , Amer. Math. Soc.  (1979)  pp. 427–430</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  C. Fefferman,  "Characterization of bounded mean oscillation"  ''Bull. Amer. Math. Soc.'' , '''77'''  (1971)  pp. 587–588</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  C. Fefferman,  E. Stein,  "$H ^ { p }$ spaces of several variables"  ''Acta Math.'' , '''129'''  (1974)  pp. 137–193</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  W. Hayman,  Ch. Pommerenke,  "On analytic functions of bounded mean oscillation"  ''Bull. London Math. Soc.'' , '''10'''  (1978)  pp. 219–224</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  F. John,  L. Nirenberg,  "On functions of bounded mean oscillation"  ''Commun. Pure Appl. Math.'' , '''14'''  (1961)  pp. 415–426</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  Ch. Pommerenke,  "Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation"  ''Comment. Math. Helv.'' , '''152'''  (1977)  pp. 591–602</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  K. Stephenson,  D. Stegenga,  "A geometric characterization of analytic functions of bounded mean oscillation"  ''J. London Math. Soc. (2)'' , '''24'''  (1981)  pp. 243–254</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  D. Stegenga,  "A geometric condition that implies BMOA" , ''Proc. Symp. Pure Math.'' , '''XXXV:1''' , Amer. Math. Soc.  (1979)  pp. 427–430</td></tr></table>

Revision as of 16:55, 1 July 2020

space of analytic functions of bounded mean oscillation

In 1961, F. John and L. Nirenberg [a4] introduced the space of functions of bounded mean oscillation, $\operatorname{BMO}$, in their study of differential equations (cf. also $\operatorname{BMO}$-space). About a decade later, C. Fefferman proved his famous duality theorem [a1] [a2], which states that the dual of the Hardy space $H ^ { 1 }$ is (cf. also Hardy spaces).

In these early works, $\operatorname{BMO}$ was studied primarily as a space of real-valued functions, but Fefferman's result raised questions about the nature of analytic functions in the Hardy spaces of the unit disc whose boundary values are in $\operatorname{BMO}$. This is the definition of and the duality theorem provides the alternative that consists of those analytic functions that can be represented as a sum of two analytic functions, one with a bounded real part and the other with a bounded imaginary part (cf. also Analytic function; Hardy spaces).

Ch. Pommerenke [a5] proved that a univalent function is in if and only if there is a bound on the radius of the discs contained in the image. Subsequently, W. Hayman and Pommerenke [a3] and D. Stegenga [a7] proved that any analytic function is in provided the complement of its image is sufficiently thick in a technical sense that uses the notion of logarithmic capacity. As an example, any function whose image does not contain a disc of a fixed radius $\epsilon > 0$ centred at $a + i b$, where $a$, $b$ range over all integers, satisfies this criterion and hence is in .

In a similar vein, K. Stephenson and Stegenga [a6] proved that an analytic function is in provided its image Riemann surface (viewed as spread out over the complex plane) has the following property: There are $0 < R < \infty$, $0 < \epsilon < 1$ so that a Brownian traveller will, with probability at least , fall off the edge of the surface before travelling outward $R$ units (cf. also Brownian motion). As an example, an overlapping infinite saussage-shaped region can be constructed so that the Riemann mapping function maps onto to the entire complex plane but is nevertheless in .

comes up naturally in many problems in analysis, such as on the composition operator, the corona problem (cf. also Hardy classes), and on functions in one and several complex variables. $\operatorname{BMO}$ and its variants has become an indispensable tool in real and complex analysis.

References

[a1] C. Fefferman, "Characterization of bounded mean oscillation" Bull. Amer. Math. Soc. , 77 (1971) pp. 587–588
[a2] C. Fefferman, E. Stein, "$H ^ { p }$ spaces of several variables" Acta Math. , 129 (1974) pp. 137–193
[a3] W. Hayman, Ch. Pommerenke, "On analytic functions of bounded mean oscillation" Bull. London Math. Soc. , 10 (1978) pp. 219–224
[a4] F. John, L. Nirenberg, "On functions of bounded mean oscillation" Commun. Pure Appl. Math. , 14 (1961) pp. 415–426
[a5] Ch. Pommerenke, "Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation" Comment. Math. Helv. , 152 (1977) pp. 591–602
[a6] K. Stephenson, D. Stegenga, "A geometric characterization of analytic functions of bounded mean oscillation" J. London Math. Soc. (2) , 24 (1981) pp. 243–254
[a7] D. Stegenga, "A geometric condition that implies BMOA" , Proc. Symp. Pure Math. , XXXV:1 , Amer. Math. Soc. (1979) pp. 427–430
How to Cite This Entry:
BMOA-space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=BMOA-space&oldid=17734
This article was adapted from an original article by D. Stegenga (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article