CHAPTER 0

Overview

The theory of Bergman spaces has evolved from several sources. A

primary model is the related theory of Hardy spaces. For 0 p oo, a

function / analytic in the unit disk D is said to belong to the Hardy space

HP if the integrals /^

\f{rel0)\pd6

remain bounded as r — 1. It belongs to

the Bergman space

Ap

if the area integral J

p

\f(z)\pda

is finite. It is clear

that

Hp

C

Ap.

The space H°° consists of the bounded analytic functions.

The structural properties of individual functions in

Hp

were studied

actively in the period 1915-1930, beginning with a classical paper by

G. H. Hardy. It was found that each such function has radial limits at

almost every point of the unit circle, and that the boundary function

cannot vanish on any set of positive measure unless the function vanishes

identically. The zero-sets {z^} of functions in

Hp

were neatly characterized

by the Blaschke condition ^ ( 1 — |^|). oo, a property independent of p.

For 1 p oo the space

Hp

was shown to be invariant under harmonic

conjugation. Detailed information was obtained about Taylor coefficients.

One early impetus for the study of Hardy spaces was their application to

the theory of trigonometric series.

With the emergence of functional analysis in the 1930s,

Hp

spaces began

to be viewed as examples of Banach spaces, for 1 p oo. This point of

view suggested a variety of new problems and provided effective methods

for the solution of old problems. In 1949, a seminal paper of Beurling [2]

gave a complete description of the invariant subspaces of the shift operator

in the sequence space

£2

by relating it to the operator of multiplication

by z in the Hardy space

H2.

Beurling coined the terms "inner function"

and "outer function" and made essential use of the canonical factorization

of an arbitrary

H2

function into a product of inner and outer functions,

a result due to Smirnov. At about the same time, S. Ya. Havinson and

H. S. Shapiro independently introduced a theory of dual extremal problems

in

Hp

spaces, unifying an existing collection of examples into a coherent

framework that could be fully understood only in the setting of functional

analysis. Around 1960, Lennart Carleson [2,3] and later Shapiro and Shields

[1] solved universal interpolation problems in

Hp

spaces with the help of

functional analysis. A prominent feature of these developments was the

interplay of "hard" and "soft" analysis that has given the subject a special

appeal.

1

http://dx.doi.org/10.1090/surv/100/01