# 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 $\operatorname{BMOA}$ (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 $\operatorname{BMOA}$ and the duality theorem provides the alternative that $\operatorname{BMOA}$ 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 $\operatorname{BMOA}$ 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 $\operatorname{BMOA}$ 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 $\operatorname{BMOA}$.

In a similar vein, K. Stephenson and Stegenga [a6] proved that an analytic function is in $\operatorname{BMOA}$ 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 $\varepsilon$, 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 $\operatorname{BMOA}$.

$\operatorname{BMOA}$ 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 |

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BMOA-space.

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