A Characterization of Random Bloch Functions

In this paper, we introduce a necessary and sufficient condition on the complex sequence {an}, ∑ |an| < ∞, so that ∞n=1±anzn represents a Bloch function for almost all choices of signs “±”, answering a question left open in [2]. AMS 1991 Subject Classifications: Primary: 30B20; secondary: 60G15. Introduction A Bloch function is an analytic function f(z) in the unit disk D = {z : |z| < 1}, such that sup z∈D (1− |z|2)|f ′(z)| < ∞. When equipped with the norm ‖f‖B = |f(0)|+ sup z∈D (1− |z|2)|f ′(z)|, the set of all Bloch functions forms a Banach space, called the Bloch space. In this note, we study the random power series fω(z) = ∞ ∑ n=0 anεn(ω)z n where {εn(ω)} is a Rademacher sequence, that is εn = ±1. In particular, we will consider the following problem raised by Anderson in [1]: Problem Find a necessary and sufficient condition on {an}, such that for Rademacher sequence {εn(ω)}, the series fω(z) = ∞ ∑ n=0 anεn(ω)z n represents a Bloch function almost surely. For the history and the related research, see e.g. [2], [3] and [1]. 1 The study of random series dates back at least to Paley and Zygmund (1930). For a long time, a major question was to characterize the a.s. convergence of the random Fourier series ∞ ∑ n=0 anεne , where {an} is a sequence of numbers satisfying ∞n=0 |an| < ∞. This question was completely solved by Marcus and Pisier ([4]). Their result will be adapted in this paper to produce the proof of the sufficient part of the following theorem. Theorem 1 If {εn} is a Rademacher sequence, then the random power series fω(z) = ∞ ∑ n=0 anεn(ω)z n is a Bloch function almost surely if and only if ∫ ∞ 0 dn(e −t)dt = O(n), where dn is the non-decreasing rearrangement of dn(t) = √√√√ n ∑ k=1 k2|ak|2|e2πkti − 1|2. Here and throughout this note, the non-decreasing rearrangement of a (Lebesgue) mmeasurable function h(t) on [0, 1] is defined by h(s) = sup{y : m({t : h(t) < y}) < s}. Marcus-Pisier In this section, we introduce a result of Marcus and Pisier [4]. For the notational simplicity, we define ρ(t) to be the non-decreasing rearrangement of