Besov-Lipschitz and mean Besov-Lipschitz spaces of holomorphic functions on the unit ball. (English) Zbl 1266.32004
Potential Anal. 38, No. 4, 1187-1206 (2013); addendum ibid. 41, No. 1, 31-34 (2014).
Summary: We give several characterizations of holomorphic mean Besov-Lipschitz spaces on the unit ball in \({\mathbb C^N} \) and appropriate Besov-Lipschitz spaces and prove equivalences between them. Equivalent norms on the mean Besov-Lipschitz spaces involve different types of \(L ^{p }\)-moduli of continuity, while in the characterizations of Hardy-Sobolev spaces we use not only the radial derivative but also the gradient. A characterization in terms of the best approximation by polynomials is also given.
MSC:
| 32A35 | \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables |
| 32A36 | Bergman spaces of functions in several complex variables |
| 32A37 | Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) |
References:
| [1] | Ahern, P., Bruna, J.: Besov spaces, Sobolev spaces, and Cauchy integrals. Mich. Math. J. 39, 239-261 (1992) · Zbl 0767.46022 · doi:10.1307/mmj/1029004520 |
| [2] | Ahern, P., Schneider, R.: Holomorphic Lipschitz functions in pseudoconvex domains. Am. J. Math. 101, 543-565 (1979) · Zbl 0455.32008 · doi:10.2307/2373797 |
| [3] | Cho, H.R., Zhu, K.: Holomorphic mean Lipschitz spaces and Hardy Sobolev spaces on the unit ball. To appear in Complex Variables · Zbl 1252.32009 |
| [4] | Cho, H.R., Koo, H., Kwon, E.G.: Holomorphic mean Lipschitz spaces and Besov spaces on the unit ball in \(\mathbb C^N\). To appear in J. Korean Math. Soc. · Zbl 1300.32003 |
| [5] | Duren, P.L.: Theory of Hp Spaces. Academic Press, New York (1970). Reprinted with supplement by Dover Publications, Mineola, NY (2000) |
| [6] | Hardy, G.H., Littlewood, J.E.: Some properties of fractional integrals. I. Math. Z. 27(1), 565-606 (1928) · JFM 54.0275.05 · doi:10.1007/BF01171116 |
| [7] | Jevtić, M., Pavlović, M.: On the Hahn-Banach extension property in Hardy and mixed norm spaces on the unit ball. Monatsh. Math. 111(2), 137-145 (1991) · Zbl 0722.32006 · doi:10.1007/BF01332352 |
| [8] | Jevtić, M., Pavlović, M.: Coefficient multipliers on spaces of analytic functions. Acta Sci. Math. (Szeged) 64, 531-545 (1998) · Zbl 0926.46022 |
| [9] | Mateljević, M., Pavlović, M.: Lp behaviour of the integral means of analytic functions. Stud. Math. 77, 219-237 (1984) · Zbl 1188.30004 |
| [10] | Mateljević, M., Pavlović, M.: The best approximation and composition with inner functions. Mich. Math. J. 42(2), 367-378 (1995) · Zbl 0844.41019 · doi:10.1307/mmj/1029005234 |
| [11] | Nikiforov, A.F., Uvarov, V.B.: Special Functions of Mathematical Physics. A Unified Introduction with Applications, vol. XVIII, 427 pp. Birkhäuser. Basel-Boston (1988). Transl. from the Russian by Ralph P. Boas · Zbl 0624.33001 |
| [12] | Oswald, P.: On Besov-Hardy-Sobolev spaces of analytic functions in the unit disc. Czech. Math. J. 33, 408-426 (1983) · Zbl 0594.46027 |
| [13] | Oswald, P.: On the moduli of continuity of Hp functions with 0 < p < 1. Proc. Edinb. Math. Soc. 35, 89-100 (1992) · Zbl 0754.30029 · doi:10.1017/S0013091500005344 |
| [14] | Oswald, P.: Decompositions of Lp and Hardy spaces of polyharmonic functions. J. Math. Anal Appl. 216, 499-509 (1997) · Zbl 0912.46032 · doi:10.1006/jmaa.1997.5675 |
| [15] | Oswald, P., Introduction to function spaces on the disk (2004), Belgrade · Zbl 1107.30001 |
| [16] | Rudin, W.: Function Theory in the Unit Ball in \(\mathbb C^n\). Springer, New York (1980) · Zbl 0495.32001 · doi:10.1007/978-1-4613-8098-6 |
| [17] | Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball. Springer, New York (2005) · Zbl 1067.32005 |
| [18] | Zygmund, A.: Smooth functions. Duke Math. J. 12, 47-76 (1945) · Zbl 0060.13806 · doi:10.1215/S0012-7094-45-01206-3 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
