Hadamard gap series in growth spaces

TL;DR

This paper characterizes harmonic functions in growth spaces represented by Hadamard gap series, linking their coefficients to growth and oscillation properties, and applies probabilistic methods to establish sharp bounds.

Contribution

It provides a new characterization of Hadamard gap series in harmonic growth spaces with doubling weights, connecting coefficient conditions to growth and oscillation behaviors.

Findings

Functions grow slower than their majorant or oscillate along most radii.

Characterization of Hadamard gap series in Bloch-type spaces with doubling weights.

Established sharp upper bounds on growth using the law of the iterated logarithm.

Abstract

Let $h^\infty_v$ 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 $h^\infty_v$ 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 $u\in h^\infty_v$ 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.