Abstract
We consider harmonic Bergman-Besov spaces \(b^p_{\alpha}\) and weighted Bloch spaces \(b^{\infty}_\alpha\) on the unit ball of ℝn 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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Axler, P. Bourdon, W. Ramey: Harmonic Function Theory. Graduate Texts in Mathematics 137. Springer, New York, 2001.
B. R. Choe, H. Koo, Y. Lee: Positive Schatten class Toeplitz operators on the ball. Studia Math. 189 (2008), 65–90.
B. R. Choe, Y. J. Lee: Note on atomic decompositions of harmonic Bergman functions. Complex Analysis and Its Applications, OCAMI Studies 2 (Imayoshi Yoichi etal., eds.). Osaka Municipal Universities Press, Osaka, 2007, pp. 11–24.
B. R. Choe, Y. J. Lee, K. Na: Positive Toeplitz operators from a harmonic Bergman space into another. Tohoku Math. J. 56 (2004), 255–270.
B. R. Choe, Y. J. Lee, K. Na: Toeplitz operators on harmonic Bergman spaces. Nagoya Math. J. 174 (2004), 165–186.
R. R. Coifman, R. Rochberg: Representation theorems for holomorphic and harmonic functions in L p. Astérisque 77 (1980), 11–66.
A. E. Djrbashian, F. A. Shamoian: Topics in the Theory of A pα Spaces. Teubner Texts in Mathematics, 105, B. G. Teubner, Leipzig, 1988.
Ö. F. Doğan: Harmonic Besov spaces with small exponents. Available at https://doi.org/abs/1808.01451.
Ö. F. Doğan, A. E. Üreyen: Weighted harmonic Bloch spaces on the ball. Complex Anal. Oper. Theory 12 (2018), 1143–1177.
E. Doubtsov: Carleson-Sobolev measures for weighted Bloch spaces. J. Funct. Anal. 258 (2010), 2801–2816.
S. Gergün, H. T. Kaptanoğlu, A. E. Üreyen: Reproducing kernels for harmonic Besov spaces on the ball. C. R. Math. Acad. Sci. Paris 347 (2009), 735–738.
S. Gergün, H. T. Kaptanoğlu, A. E. Üreyen: Harmonic Besov spaces on the ball. Int. J. Math. 27 (2016), Article ID 1650070, 59 pages.
M. Jevtić, M. Pavlović: Harmonic Bergman functions on the unit ball in ℝn. Acta Math. Hung. 85 (1999), 81–96.
C. W. Liu, J. H. Shi: Invariant mean-value property and \({\cal M}\)-harmonicity in the unit ball of ℝn. Acta Math. Sin. 19 (2003), 187–200.
D. H. Luecking: Multipliers of Bergman spaces into Lebesgue spaces. Proc. Edinburgh Math. Soc. 29 (1986), 125–131.
D. H. Luecking: Embedding theorems for spaces of analytic functions via Khinchine’s inequality. Michigan Math. J. 40 (1993), 333–358.
J. Miao: Reproducing kernels for harmonic Bergman spaces of the unit ball. Monatsh. Math. 125 (1998), 25–35.
V. L. Oleinik, B. S. Pavlov: Embedding theorems for weighted classes of harmonic and analytic functions. J. Soviet Math. 2 (1974), 135–142 (In English. Russian original.); translation from Zap. Nauch. Sem. LOMI Steklov 22 (1971), 94–102.
G. Ren: Harmonic Bergman spaces with small exponents in the unit ball. Collect. Math. 53 (2002), 83–98.
W. Yang, C. Ouyang: Exact location of ∝-Bloch spaces in L pα and H p of a complex unit ball. Rocky Mt. J. Math. 30 (2000), 1151–1169.
R. Zhao, K. Zhu: Theory of Bergman spaces in the unit ball of ℂn. Mém. Soc. Math. Fr. 115 (2008), 103 pages.
A. Zygmund: Trigonometric Series. Vol. I, II. Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2002.
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Doğan, Ö.F., Üreyen, A.E. Inclusion relations between harmonic Bergman-Besov and weighted Bloch spaces on the unit ball. Czech Math J 69, 503–523 (2019). https://doi.org/10.21136/CMJ.2018.0422-17
Received:
Published:
Issue Date:
DOI: https://doi.org/10.21136/CMJ.2018.0422-17