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
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