Inclusion relations between harmonic Bergman-Besov and weighted Bloch spaces on the unit ball

Ömer Faruk Doğan; Adem Ersin Üreyen

Czechoslovak Mathematical Journal (2019)

  • Volume: 69, Issue: 2, page 503-523
  • ISSN: 0011-4642

Abstract

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We consider harmonic Bergman-Besov spaces b α p and weighted Bloch spaces b α ∞ on the unit ball of ℝ n for the full ranges of parameters 0 < p < ∞ , α ∈ 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 α > 0 .

How to cite

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Doğan, Ömer Faruk, and Üreyen, Adem Ersin. "Inclusion relations between harmonic Bergman-Besov and weighted Bloch spaces on the unit ball." Czechoslovak Mathematical Journal 69.2 (2019): 503-523. <http://eudml.org/doc/294210>.

@article{Doğan2019,
abstract = {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 $\alpha >0$.},
author = {Doğan, Ömer Faruk, Üreyen, Adem Ersin},
journal = {Czechoslovak Mathematical Journal},
keywords = {harmonic Bergman-Besov space; weighted harmonic Bloch space; Carleson measure; Berezin transform},
language = {eng},
number = {2},
pages = {503-523},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Inclusion relations between harmonic Bergman-Besov and weighted Bloch spaces on the unit ball},
url = {http://eudml.org/doc/294210},
volume = {69},
year = {2019},
}

TY - JOUR
AU - Doğan, Ömer Faruk
AU - Üreyen, Adem Ersin
TI - Inclusion relations between harmonic Bergman-Besov and weighted Bloch spaces on the unit ball
JO - Czechoslovak Mathematical Journal
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 2
SP - 503
EP - 523
AB - 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 $\alpha >0$.
LA - eng
KW - harmonic Bergman-Besov space; weighted harmonic Bloch space; Carleson measure; Berezin transform
UR - http://eudml.org/doc/294210
ER -

References

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  1. Axler, S., Bourdon, P., Ramey, W., 10.1007/b97238, Springer, New York (2001). (2001) Zbl0959.31001MR1805196DOI10.1007/b97238
  2. Choe, B. R., Koo, H., Lee, Y., 10.4064/sm189-1-6, Studia Math. 189 (2008), 65-90. (2008) Zbl1155.47030MR2443376DOI10.4064/sm189-1-6
  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. (2007) Zbl1154.47019MR2405697
  4. Choe, B. R., Lee, Y. J., Na, K., 10.2748/tmj/1113246553, Tohoku Math. J. 56 (2004), 255-270. (2004) Zbl1077.47028MR2053321DOI10.2748/tmj/1113246553
  5. Choe, B. R., Lee, Y. J., Na, K., 10.1017/S0027763000008837, Nagoya Math. J. 174 (2004), 165-186. (2004) Zbl1067.47039MR2066107DOI10.1017/S0027763000008837
  6. Coifman, R. R., Rochberg, R., Representation theorems for holomorphic and harmonic functions in L p , Astérisque 77 (1980), 11-66. (1980) Zbl0472.46040MR0604369
  7. Djrbashian, A. E., Shamoian, F. A., Topics in the Theory of A α p Spaces, Teubner Texts in Mathematics, 105, B. G. Teubner, Leipzig (1988). (1988) Zbl0667.30032MR1021691
  8. Doğan, "{O}. F., Harmonic Besov spaces with small exponents, Available at https://arxiv.org/abs/1808.01451. 
  9. Doğan, Ö. F., Üreyen, A. E., 10.1007/s11785-017-0645-9, Complex Anal. Oper. Theory 12 (2018), 1143-1177. (2018) Zbl06909437MR3800965DOI10.1007/s11785-017-0645-9
  10. Doubtsov, E., 10.1016/j.jfa.2009.10.028, J. Funct. Anal. 258 (2010), 2801-2816. (2010) Zbl1191.32003MR2593344DOI10.1016/j.jfa.2009.10.028
  11. Gergün, S., Kaptanoğlu, H. T., Üreyen, A. E., 10.1016/j.crma.2009.04.016, C. R. Math. Acad. Sci. Paris 347 (2009), 735-738. (2009) Zbl1179.31003MR2543973DOI10.1016/j.crma.2009.04.016
  12. Gergün, S., Kaptanoğlu, H. T., Üreyen, A. E., 10.1142/S0129167X16500701, Int. J. Math. 27 (2016), Article ID 1650070, 59 pages. (2016) Zbl1354.31005MR3546608DOI10.1142/S0129167X16500701
  13. Jevtić, M., Pavlović, M., 10.1023/A:1006620929091, Acta Math. Hung. 85 (1999), 81-96. (1999) Zbl0956.32004MR1713093DOI10.1023/A:1006620929091
  14. Liu, C. W., Shi, J. H., 10.1007/s10114-002-0203-9, Acta Math. Sin. 19 (2003), 187-200. (2003) Zbl1031.31001MR1968481DOI10.1007/s10114-002-0203-9
  15. Luecking, D. H., 10.1017/S001309150001748X, Proc. Edinburgh Math. Soc. 29 (1986), 125-131. (1986) Zbl0587.30048MR0829188DOI10.1017/S001309150001748X
  16. Luecking, D. H., 10.1307/mmj/1029004756, Michigan Math. J. 40 (1993), 333-358. (1993) Zbl0801.46019MR1226835DOI10.1307/mmj/1029004756
  17. Miao, J., 10.1007/BF01489456, Monatsh. Math. 125 (1998), 25-35. (1998) Zbl0907.46020MR1485975DOI10.1007/BF01489456
  18. Oleinik, V. L., Pavlov, B. S., 10.1007/BF01099672, J. Soviet Math. 2 (1974), 135-142 English. Russian original translation from Zap. Nauch. Sem. LOMI Steklov 22 1971 94-102. (1974) Zbl0278.46032MR0318867DOI10.1007/BF01099672
  19. Ren, G., Harmonic Bergman spaces with small exponents in the unit ball, Collect. Math. 53 (2002), 83-98. (2002) Zbl1029.46019MR1893309
  20. Yang, W., Ouyang, C., 10.1216/rmjm/1021477265, Rocky Mt. J. Math. 30 (2000), 1151-1169. (2000) Zbl0978.32002MR1797836DOI10.1216/rmjm/1021477265
  21. Zhao, R., Zhu, K., 0.24033/msmf.427, Mém. Soc. Math. Fr. 115 (2008), 103 pages. (2008) Zbl1176.32001MR2537698DOI0.24033/msmf.427
  22. Zygmund, A., Trigonometric Series. Vol. I, II, Cambridge Mathematical Library, Cambridge University Press, Cambridge (2002). (2002) Zbl1084.42003MR1963498

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