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.
Similar content being viewed by others
References
Abakumov E., Doubtsov E.: Reverse estimates in growth spaces. Math. Z. 271, 399–413 (2012)
Bennett G., Stegenga D.A., Timoney R.M.: Coefficients of Bloch and Lipschitz functions. Ill. J. Math. 25, 520–531 (1981)
Borichev A.: On the minimum of harmonic functions. J. Anal. Math. 89, 199–212 (2003)
Borichev A., Lyubarskii Yu., Malinnikova E., Thomas P.: Radial growth of functions in Korenblum space. Algebra i Analiz 21, 47–65 (2009)
Cartwright M.L.: On analytic functions regular in the unit circle. Q. J. Math. Oxf. Ser. 4, 246–257 (1933)
Eikrem, K.S., Malinnikova, E.: Radial growth of harmonic functions in the unit ball. Math. Scand. (to appear)
Kahane J.-P., Weiss M., Weiss G.: On lacunary power series. Ark. Mat. 5, 1–26 (1963)
Korenblum B.: An extension of the Nevanlinna theory. Acta Math. 135, 187–219 (1975)
Kwon E.G., Pavlović M.: Bibloch mappings and composition operators from Bloch type spaces to BMOA. J. Math. Anal. Appl. 382(1), 303–313 (2011)
Lyubarskii Yu., Malinnikova E.: Radial oscillation of harmonic functions in the Korenblum class. Bull. Lond. Math. Soc. 44(1), 68–84 (2012)
Nikolski, N.K.: Selected problems of weighted approximation and spectral analysis. American Mathematical Society, New York (1976)
Pavlović M.: Lacunary series in weighted spaces of analytic functions. Arch. Math. 97(5), 467–473 (2011)
Rubel L.A., Shields A.L.: The second duals of certain spaces of analytic functions. J. Aust. Math. Soc. 11, 276–280 (1970)
Salem R., Zygmund A.: La loi du logarithme iteré pour les séries trigonométriques lacunaires. Bull. Sci. Math. II Ser. 74, 209–224 (1950)
Shields A.L, Williams D.L.: Bounded projections, duality, and multipliers in spaces of harmonic functions. J. Reine. Angew. Math. 299–300, 256–279 (1978)
Stević S.: Bloch-type functions with Hadamard gaps. Appl. Math. Comput. 208(2), 416–422 (2009)
Stroock, D.W.: A concise introduction to the theory of integration, 3rd edn. Birkhäuser, Boston (1999)
Weiss M.: The law of the iterated logarithm for lacunary trigonometric series. Trans. Am. Math. Soc. 91, 444–469 (1959)
Wulan H., Zhu K.: Lacunary series in Q K spaces. Stud. Math. 178(3), 217–230 (2007)
Yamashita S.: Gap series and alpha-Bloch functions. Yokohama Math. J. 28, 31–36 (1980)
Yang C., Xu W.: Spaces with normal weights and Hadamard gap series. Arch. Math. 96(2), 151–160 (2011)
Zygmund A.: Trigonometric series. Cambridge University Press, Cambridge (1968)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Eikrem, K.S. Hadamard gap series in growth spaces. Collect. Math. 64, 1–15 (2013). https://doi.org/10.1007/s13348-012-0065-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13348-012-0065-0