Hadamard gap series in growth spaces

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.