View source for BMOA-space

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''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, $\operatorname{BMO}$, 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, $\operatorname{BMO}$ 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 $\epsilon &gt; 0$ 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 $0 &lt; R &lt; \infty$, $0 &lt; \epsilon &lt; 1$ 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. $\operatorname{BMO}$ and its variants has become an indispensable tool in real and complex analysis.

====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,   "$H ^ { p }$ 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>