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
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