Hadamard gap series in growth spaces

Abstract

Abstract. Let h ∞ v be the class of harmonic functions in the unit disk which admit a two-sided radial majorant v(r). We consider functions v that fulfill a doubling condition. We characterize functions in h ∞ v that are represented by Hadamard gap series in terms of their coefficients, and as a corollary we obtain a characterization of Hadamard gap series in Bloch-type spaces for weights with a doubling property. We show that if u ∈ h ∞ v is represented by a Hadamard gap series, then u will grow slower than v or oscillate along almost all radii. We use the law of the iterated logarithm for trigonometric series to find an upper bound on the growth of a weighted average of the function u, and we show that the estimate is sharp.1. Introduction 1.1. Growth spaces of harmonic functions. Let v be a positive increasing continuous function on [0, 1), assume that v(0) = 1 and lim r→1 v(r) = +∞. We study the growth spaces of harmonic functions in the unit disk:(1) h ∞ v = {u : D → R, ∆u = 0, |u(z)| ≤ Kv(|z|) for some K > 0}. In this article we mainly deal with weights which satisfy the doubling conditionFor such weights we characterize functions in h ∞ v represented by Hadamard gap series. Further, using this characterization, we study radial behavior of such functions in more detail.Growth spaces appear naturally as duals to the classical spaces of harmonic and analytic functions; a similar class of harmonic functions satisfying only one-sided estimate with v(r) = log 1 1−r was introduced by B. Korenblum in [8]. General growth spaces can be found in the works of L. Rubel and A. Shields, and A. Shields and D. Williams, see [11,13]. Multidimensional analogs were recently considered in [1,6]. Various results on coefficients of functions in growth spaces were obtained in [2].1.2. Hadamard gap series. Hadamard gap series are functions of the formwhere n k+1 > λn k and λ > 1; for the harmonic case we take u = ℜf . For various spaces of analytic and harmonic functions Hadamard gap series in these spaces can be characterized in terms of the coefficients a n . For example, α-Bloch spaces are 2010 Mathematics Subject Classification. Primary: 31A20; Secondary: 42A55, 60F15.

Random Harmonic Functions in Growth Spaces and Bloch-type Spaces

“…is sufficient to imply that u ∈ h ∞ v (D), but it is not necessary. In the special case of the Hadamard gap series, (1.6) is both necessary and sufficient; see [5], and this is also the case when all the coefficients are positive [4]. But it is not possible in general to characterize all functions in h ∞ v (D) by the absolute value of their coefficients.…”

“…Various results on the coefficients of functions in growth spaces were obtained in [4]. Hadamard gap series in growth spaces have been studied by a number of authors; see [5] and references therein.…”

“…Examples of functions in h ∞ v (D) can be constructed by lacunary series; see [5]. Another way to construct examples is by using random series, and such functions will be the main focus of this paper.…”

“…Then almost surely u(z, ξ) = ∞ j=0 a j ξ j r j cos jθ / ∈ h ∞ v (D). 1); see [5]. But all the coefficients cannot grow this fast if u ∈ h ∞ v (D):…”

Characterisations of Hardy Growth Spaces With Doubling Weights

“…The doubling property (1.2) is a natural technical assumption (see, for example, [1,[3][4][5]). In particular, Ω α with α > 0, the normal weights and ω β with β > 0 are doubling weights.…”