CHAPTER 1 Basic Notation and Introduction to Weights In this chapter we first define the weighted Bergman spaces and the classical Hardy spaces of the unit disc and fix the basic notation. Then we introduce the classes of radial and non-radial weights that are considered in the monograph, show relations between them, and prove several lemmas on weights that are instrumental for the rest of the monograph. 1.1. Basic notation Let H(D) denote the algebra of all analytic functions in the unit disc D = {z : |z| 1} of the complex plane C. Let T be the boundary of D, and let D(a, r) = {z : |z a| r} denote the Euclidean disc of center a C and radius r (0, ∞). A function ω : D (0, ∞), integrable over D, is called a weight function or simply a weight. It is radial if ω(z) = ω(|z|) for all z D. For 0 p and a weight ω, the weighted Bergman space p consists of those f H(D) for which f p p = D |f(z)|pω(z) dA(z) ∞, where dA(z) = dx dy π is the normalized Lebesgue area measure on D. As usual, we write Ap α for the classical weighted Bergman space induced by the standard radial weight ω(z) = (1 |z|2)α, −1 α ∞. For 0 p ∞, the Hardy space Hp consists of those f H(D) for which f Hp = lim r→1− Mp(r, f) ∞, where Mp(r, f) = 1 0 |f(reiθ)|p 1 p , 0 p ∞, and M∞(r, f) = max 0≤θ≤2π |f(reiθ)|. For the theory of the Hardy and the classical weighted Bergman spaces, see [28, 29, 31, 42, 87]. Throughout the monograph, the letter C = C(·), with a subscript if needed, will denote an absolute constant whose value depends on the parameters indicated in the parenthesis, and may change from one occurrence to another. We will use the notation a b if there exists a constant C = C(·) 0 such that a Cb, and a b is understood in an analogous manner. In particular, if a b and a b, then we will write a b. 5
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