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.