Hadamard gap series in growth spaces

• Autores: Kjersti Solberg Eikrem
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 64, Fasc. 1, 2013, págs. 1-15
• Idioma: inglés
• DOI: 10.1007/s13348-012-0065-0
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• Resumen

  • Let ℎ∞? 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∞? 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 ?∈ℎ∞? 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.