# BMOA-space

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.

How to Cite This Entry:
BMOA-space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=BMOA-space&oldid=50586
This article was adapted from an original article by D. Stegenga (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article