| Title:
|
Inclusion relations between harmonic Bergman-Besov and weighted Bloch spaces on the unit ball (English) |
| Author:
|
Doğan, Ömer Faruk |
| Author:
|
Üreyen, Adem Ersin |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
69 |
| Issue:
|
2 |
| Year:
|
2019 |
| Pages:
|
503-523 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We consider harmonic Bergman-Besov spaces $b^p_\alpha $ and weighted Bloch spaces $b^\infty _\alpha $ on the unit ball of $\mathbb {R}^n$ for the full ranges of parameters $0<p<\infty $, $\alpha \in \mathbb {R}$, and determine the precise inclusion relations among them. To verify these relations we use Carleson measures and suitable radial differential operators. For harmonic Bergman spaces various characterizations of Carleson measures are known. For weighted Bloch spaces we provide a characterization when $\alpha >0$. (English) |
| Keyword:
|
harmonic Bergman-Besov space |
| Keyword:
|
weighted harmonic Bloch space |
| Keyword:
|
Carleson measure |
| Keyword:
|
Berezin transform |
| MSC:
|
31B05 |
| MSC:
|
42B35 |
| idZBL:
|
Zbl 07088802 |
| idMR:
|
MR3959962 |
| DOI:
|
10.21136/CMJ.2018.0422-17 |
| . |
| Date available:
|
2019-05-24T09:01:43Z |
| Last updated:
|
2021-07-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147742 |
| . |
| Reference:
|
[1] Axler, S., Bourdon, P., Ramey, W.: Harmonic Function Theory. Graduate Texts in Mathematics 137.Springer, New York (2001). Zbl 0959.31001, MR 1805196, 10.1007/b97238 |
| Reference:
|
[2] Choe, B. R., Koo, H., Lee, Y.: Positive Schatten class Toeplitz operators on the ball.Studia Math. 189 (2008), 65-90. Zbl 1155.47030, MR 2443376, 10.4064/sm189-1-6 |
| Reference:
|
[3] Choe, B. R., Lee, Y. J.: Note on atomic decompositions of harmonic Bergman functions.Complex Analysis and Its Applications, OCAMI Studies 2 Imayoshi Yoichi et al. Osaka Municipal Universities Press, Osaka (2007), 11-24. Zbl 1154.47019, MR 2405697 |
| Reference:
|
[4] Choe, B. R., Lee, Y. J., Na, K.: Positive Toeplitz operators from a harmonic Bergman space into another.Tohoku Math. J. 56 (2004), 255-270. Zbl 1077.47028, MR 2053321, 10.2748/tmj/1113246553 |
| Reference:
|
[5] Choe, B. R., Lee, Y. J., Na, K.: Toeplitz operators on harmonic Bergman spaces.Nagoya Math. J. 174 (2004), 165-186. Zbl 1067.47039, MR 2066107, 10.1017/S0027763000008837 |
| Reference:
|
[6] Coifman, R. R., Rochberg, R.: Representation theorems for holomorphic and harmonic functions in $L^p$.Astérisque 77 (1980), 11-66. Zbl 0472.46040, MR 0604369 |
| Reference:
|
[7] Djrbashian, A. E., Shamoian, F. A.: Topics in the Theory of $A^p_\alpha$ Spaces.Teubner Texts in Mathematics, 105, B. G. Teubner, Leipzig (1988). Zbl 0667.30032, MR 1021691 |
| Reference:
|
[8] Doğan, "{O}. F.: Harmonic Besov spaces with small exponents.Available at https://arxiv.org/abs/1808.01451. |
| Reference:
|
[9] Doğan, Ö. F., Üreyen, A. E.: Weighted harmonic Bloch spaces on the ball.Complex Anal. Oper. Theory 12 (2018), 1143-1177. Zbl 06909437, MR 3800965, 10.1007/s11785-017-0645-9 |
| Reference:
|
[10] Doubtsov, E.: Carleson-Sobolev measures for weighted Bloch spaces.J. Funct. Anal. 258 (2010), 2801-2816. Zbl 1191.32003, MR 2593344, 10.1016/j.jfa.2009.10.028 |
| Reference:
|
[11] Gergün, S., Kaptanoğlu, H. T., Üreyen, A. E.: Reproducing kernels for harmonic Besov spaces on the ball.C. R. Math. Acad. Sci. Paris 347 (2009), 735-738. Zbl 1179.31003, MR 2543973, 10.1016/j.crma.2009.04.016 |
| Reference:
|
[12] Gergün, S., Kaptanoğlu, H. T., Üreyen, A. E.: Harmonic Besov spaces on the ball.Int. J. Math. 27 (2016), Article ID 1650070, 59 pages. Zbl 1354.31005, MR 3546608, 10.1142/S0129167X16500701 |
| Reference:
|
[13] Jevtić, M., Pavlović, M.: Harmonic Bergman functions on the unit ball in $\mathbb R^n$.Acta Math. Hung. 85 (1999), 81-96. Zbl 0956.32004, MR 1713093, 10.1023/A:1006620929091 |
| Reference:
|
[14] Liu, C. W., Shi, J. H.: Invariant mean-value property and $\mathcal M$-harmonicity in the unit ball of $\mathbb R^n$.Acta Math. Sin. 19 (2003), 187-200. Zbl 1031.31001, MR 1968481, 10.1007/s10114-002-0203-9 |
| Reference:
|
[15] Luecking, D. H.: Multipliers of Bergman spaces into Lebesgue spaces.Proc. Edinburgh Math. Soc. 29 (1986), 125-131. Zbl 0587.30048, MR 0829188, 10.1017/S001309150001748X |
| Reference:
|
[16] Luecking, D. H.: Embedding theorems for spaces of analytic functions via Khinchine's inequality.Michigan Math. J. 40 (1993), 333-358. Zbl 0801.46019, MR 1226835, 10.1307/mmj/1029004756 |
| Reference:
|
[17] Miao, J.: Reproducing kernels for harmonic Bergman spaces of the unit ball.Monatsh. Math. 125 (1998), 25-35. Zbl 0907.46020, MR 1485975, 10.1007/BF01489456 |
| Reference:
|
[18] Oleinik, V. L., Pavlov, B. S.: Embedding theorems for weighted classes of harmonic and analytic functions.J. Soviet Math. 2 (1974), 135-142 English. Russian original translation from Zap. Nauch. Sem. LOMI Steklov 22 1971 94-102. Zbl 0278.46032, MR 0318867, 10.1007/BF01099672 |
| Reference:
|
[19] Ren, G.: Harmonic Bergman spaces with small exponents in the unit ball.Collect. Math. 53 (2002), 83-98. Zbl 1029.46019, MR 1893309 |
| Reference:
|
[20] Yang, W., Ouyang, C.: Exact location of $\alpha$-Bloch spaces in $L_a^p$ and $H^p$ of a complex unit ball.Rocky Mt. J. Math. 30 (2000), 1151-1169. Zbl 0978.32002, MR 1797836, 10.1216/rmjm/1021477265 |
| Reference:
|
[21] Zhao, R., Zhu, K.: Theory of Bergman spaces in the unit ball of $\mathbb{C}^n$.Mém. Soc. Math. Fr. 115 (2008), 103 pages. Zbl 1176.32001, MR 2537698, 0.24033/msmf.427 |
| Reference:
|
[22] Zygmund, A.: Trigonometric Series. Vol. I, II.Cambridge Mathematical Library, Cambridge University Press, Cambridge (2002). Zbl 1084.42003, MR 1963498 |
| . |
