Random Harmonic Functions in Growth Spaces and Bloch-type Spaces

Eikrem, Kjersti Solberg

Canadian journal of mathematics, Tome 66 (2014) no. 2, pp. 284-302

Résumé

Let $h_{\vee }^{\infty }\,\left( \text{B} \right)$ and $h_{\vee }^{\infty }\,\left( \text{B} \right)$ be the spaces of harmonic functions in the unit disk and multidimensional unit ball admitting a two-sided radial majorant $v\left( r \right)$ . We consider functions $v$ that fulfill a doubling condition. In the two-dimensional case let $$u\left( r{{e}^{i\theta }},\xi\right)\,=\,\sum\limits_{j=0}^{\infty }{\left( {{a}_{j0}}{{\xi }_{j0}}{{r}^{j}}\,\cos \,j\theta \,+\,{{a}_{j1}}{{\xi }_{j1}}{{r}^{j}}\,\sin \,j\theta\right)}$$ where $\xi \,=\,\left\{ {{\xi }_{ji}} \right\}$ is a sequence of random subnormal variables and ${{a}_{ji}}$ are real. In higher dimensions we consider series of spherical harmonics. We will obtain conditions on the coefficients ${{a}_{ji}}$ that imply that $u$ is in $h_{\vee }^{\infty }\,\left( \text{B} \right)$ almost surely. Our estimate improves previous results by Bennett, Stegenga, and Timoney, and we prove that the estimate is sharp. The results for growth spaces can easily be applied to Bloch-type spaces, and we obtain a similar characterization for these spaces that generalizes results by Anderson, Clunie, and Pommerenke and by Guo and Liu.

Détail

DOI : 10.4153/CJM-2013-029-7

Mots-clés : 30B20, 31B05, 30H30, 42B05, harmonic functions, random series, growth space, Bloch-type space

Bibliographie

[1] [1] Abakumov, E. and Doubtsov, E., Reverse estimates in growth spaces. Math. Z. 271(2012), no. 1, 399–413. Google Scholar | DOI

[2] [2] Anderson, J. M., Clunie, J., and Pommerenke, Ch., On Bloch functions and normal functions. J. Reine Angew. Math. 270(1974), 12–37. Google Scholar

[3] [3] Avhadiev, F. G. and Kayumov, I. R. Estimates for Bloch functions and their generalization. Complex Variables Theory Appl. 29(1996), no. 3, 193–201. Google Scholar | DOI

[4] [4] Bennett, G., Stegenga, D. A., and Timoney, R. M., Coefficients of Bloch and Lipschitz functions. Illinois J. Math. 25(1981), no. 3, 520–531. Google Scholar

[5] [5] Eikrem, K. S., Hadamard gap series in growth spaces. Collect. Math. 64(2013), no. 1, 1–15. Google Scholar | DOI

[6] [6] Eikrem, K. S. and Malinnikova, E., Radial growth of harmonic functions in the unit ball. Math. Scand. 110(2012), no. 2, 273–296. Google Scholar

[7] [7] Gao, F., A characterization of random Bloch functions. J. Math. Anal. Appl. 252(2000), no. 2, 959–966. Google Scholar | DOI

[8] [8] Guo, J. and Liu, P., Random _-Bloch function. Chinese Quart. J. Math. 16(2001), no. 4, 100–103. Google Scholar

[9] [9] Kahane, J.-P., Some random series of functions. Second ed., Cambridge Studies in Advanced Mathematics, 5, Cambridge University Press, Cambridge, 1985. Google Scholar

[10] [10] Kellogg, O. D., On bounded polynomials in several variables. Math. Z. 27(1928), no. 1, 55–64. Google Scholar | DOI

[11] [11] Rubel, L. A. and Shields, A. L., The second duals of certain spaces of analytic functions. J. Aust. Math. Soc. 11(1970), no. 3, 276–280. Google Scholar | DOI

[12] [12] Rudin, W., Some theorems on Fourier coefficients. Proc. Amer. Math. Soc. 10(1959), 855–859. Google Scholar | DOI

[13] [13] Salem, R. and Zygmund, A., Some properties of trigonometric series whose terms have random signs. Acta Math. 91(1954), 245–301. Google Scholar | DOI

[14] [14] Shields, A. L. and L.Williams, D., Bounded projections, duality, and multipliers in spaces of analytic functions. Trans. Amer. Math. Soc. 162(1971), 287–302. Google Scholar

[15] [15] Shields, A. L., Bounded projections, duality, and multipliers in spaces of harmonic functions. J. Reine Angew. Math. 299/300(1978), 256–279. Google Scholar

[16] [16] Stromberg, K. R., Probability for analysts. Chapman and Hall Probability Series, Chapman & Hall, New York, 1994. Google Scholar

[17] [17] Zygmund, A., Trigonometric series. Second ed., Cambridge University Press, London-New York, 1968. Google Scholar