Preface This work concerns the weighted Bergman space Ap ω of the unit disc D that is induced by a radial continuous weight ω : [0, 1) (0, ∞) such that (‡) lim r→1− 1 r ω(s) ds ω(r)(1 r) = ∞. A radial continuous weight ω with the property (‡) is called rapidly increasing and the class of all such weights is denoted by I. Each Ap ω induced by ω I lies between the Hardy space Hp and every classical weighted Bergman space Ap α of D. In many respects the Hardy space Hp is the limit of Ap α , as α −1, but it is well known to specialists that this is a very rough estimate since none of the finer function-theoretic properties of the classical weighted Bergman space p is carried over to the Hardy space Hp whose (harmonic) analysis is much more delicate. One of the main motivations for us to write this monograph is to study the spaces Aω,p induced by rapidly increasing weights, that indeed lie “closer” to Hp than any p in the above sense and explore the change of these finer properties related to harmonic analysis. We will see that the “transition” phenomena from Hp to p does appear in the context of p with rapidly increasing weights and raises a number of new interesting problems, some of which are addressed in this monograph. Chapter 1 is devoted to proving several basic properties of the rapidly increasing weights that will be used frequently in the monograph. Also examples are provided to show that, despite of their name, rapidly increasing weights are by no means necessarily increasing and may admit a strong oscillatory behavior. Most of the presented results remain true or have analogues for those weights ω for which the quotient in the left hand side of (‡) is bounded and bounded away from zero. These weights are called regular and the class of all such weights is denoted by R. Each standard weight ω(r) = (1 r2)α is regular for all −1 α ∞. Chapter 1 is instrumental for the rest of the monograph. Many conventional tools used in the theory of the classical Bergman spaces fail to work in Ap ω that is induced by a rapidly increasing weight ω. For example, one can not find a weight ω = ω (p) such that f p f Ap ω for all analytic functions f in D with f(0) = 0, because such a Littlewood-Paley type formula does not exist unless p = 2. Moreover, neither p-Carleson measures for p can be characterized by a simple condition on pseudohyperbolic discs, nor the conjugate operator is necessarily bounded on p (of harmonic functions) if 0 p 1. However, it is shown in Chapter 2 that the embedding p Lq(μ) can be characterized by a geometric condition on Carleson squares S(I) when ω is rapidly increasing and 0 p q ∞. In these considerations we will see that the weighted maximal 1
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