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Keywords:
harmonic function; Bloch space; Besov space; majorant
harmonic function; Bloch space; Besov space; majorant
Summary:
The relationship between weighted Lipschitz functions and analytic Bloch spaces has attracted much attention. In this paper, we define harmonic $\omega $-$\alpha $-Bloch space and characterize it in terms of $$ \omega ((1-|x|^2)^\beta (1-|y|^2)^{\alpha - \beta }) \Big | \frac {f(x)-f(y)}{x-y}\Big | $$ and $$ \omega ((1-|x|^2)^\beta (1-|y|^2)^{\alpha - \beta }) \Big | \frac {f(x)-f(y)}{|x|y-x'}\Big | $$ where $\omega $ is a majorant. Similar results are extended to harmonic little $\omega $-$\alpha $-Bloch and Besov spaces. Our results are generalizations of the corresponding ones in G. Ren, U. Kähler (2005).
References:
[1] Chen, S., Ponnusamy, S., Rasila, A.: On characterizations of Bloch-type, Hardy-type and Lipschitz-type spaces. Math. Z. 279 (2015), 163-183. DOI 10.1007/s00209-014-1361-z | MR 3299847 | Zbl 1314.30116
[2] Choe, B. R., Koo, H., Yi, H.: Derivatives of harmonic Bergman and Bloch functions on the ball. J. Math. Anal. Appl. 260 (2001), 100-123. DOI 10.1006/jmaa.2000.7438 | MR 1843970 | Zbl 0984.31004
[3] Choi, E. S., Na, K.: Characterizations of the harmonic Bergman space on the ball. J. Math. Anal. Appl. 353 (2009), 375-385. DOI 10.1016/j.jmaa.2008.11.085 | MR 2508875 | Zbl 1162.31003
[4] Dyakonov, K. M.: Equivalent norms on Lipschitz-type spaces of holomorphic functions. Acta Math. 178 (1997), 143-167. DOI 10.1007/BF02392692 | MR 1459259 | Zbl 0898.30040
[5] Jevtić, M., Pavlović, M.: On $\cal M$-harmonic Bloch space. Proc. Amer. Math. Soc. 123 (1995), 1385-1392. DOI 10.2307/2161125 | MR 1264815 | Zbl 0833.32002
[6] Li, S.: Characterizations of Besov spaces in the unit ball. Bull. Korean Math. Soc. 49 (2012), 89-98. DOI 10.4134/BKMS.2012.49.1.089 | MR 2931550 | Zbl 1239.32006
[7] Li, S., Stević, S.: Some characterizations of the Besov space and the $\alpha$-Bloch space. J. Math. Anal. Appl. 346 (2008), 262-273. DOI 10.1016/j.jmaa.2008.05.044 | MR 2428290 | Zbl 1156.32002
[8] Li, S., Wulan, H.: Characterizations of $\alpha$-Bloch spaces on the unit ball. J. Math. Anal. Appl. 343 (2008), 58-63. DOI 10.1016/j.jmaa.2008.01.023 | MR 2409457 | Zbl 1204.32006
[9] Mateljević, M., Vuorinen, M.: On harmonic quasiconformal quasi-isometries. J. Inequal. Appl. 2010 (2010), Article ID 178732, 19 pages. MR 2665497 | Zbl 1211.30040
[10] Nowak, M.: Bloch space and Möbius invariant Besov spaces on the unit ball of $\mathbb C^n$. Complex Variable Theory Appl. 44 (2001), 1-12. DOI 10.1080/17476930108815339 | MR 1826712
[11] Ren, G., Kähler, U.: Weighted Lipschitz continuity and harmonic Bloch and Besov spaces in the real unit ball. Proc. Edinb. Math. Soc., II. Ser. 48 (2005), 743-755. DOI 10.1017/S0013091502000020 | MR 2171195 | Zbl 1148.42308
[12] Ren, G., Tu, C.: Bloch space in the unit ball of $\mathbb C^n$. Proc. Am. Math. Soc. 133 (2005), 719-726. DOI 10.1090/S0002-9939-04-07617-8 | MR 2113920
[13] Rudin, W.: Function Theory in the Unit Ball of $\mathbb C^n$. Grundlehren der mathematischen Wissenschaften 241 Springer, New York (1980). MR 0601594
[14] Üreyen, A. E.: An estimate of the oscillation of harmonic reproducing kernels with applications. J. Math. Anal. Appl. 434 538-553 (2016). DOI 10.1016/j.jmaa.2015.09.030 | MR 3404572 | Zbl 1358.46035
[15] Yoneda, R.: A characterization of the harmonic Bloch space and the harmonic Besov spaces by an oscillation. Proc. Edinb. Math. Soc., II. Ser. 45 (2002), 229-239. DOI 10.1017/S001309159900142X | MR 1884614 | Zbl 1032.46039
[16] Yoneda, R.: Characterizations of Bloch space and Besov spaces by oscillations. Hokkaido Math. J. 29 (2000), 409-451. DOI 10.14492/hokmj/1350912980 | MR 1776717 | Zbl 0968.46503
[17] Zhao, R.: A characterization of Bloch-type spaces on the unit ball of $\mathbb C^n$. J. Math. Anal. Appl. 330 (2007), 291-297. DOI 10.1016/j.jmaa.2006.06.100 | MR 2302923 | Zbl 1118.32006
[18] Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball. Graduate Texts in Mathematics 226 Springer, New York (2005). MR 2115155 | Zbl 1067.32005
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