×

Harmonic Besov spaces with small exponents. (English) Zbl 1436.31013

Summary: We study harmonic Besov spaces \(b_\alpha^p\) on the unit ball of \(\mathbb{R}^n\), where \(0<p<1\) and \(\alpha \in \mathbb{R}\). We provide characterizations in terms of partial and radial derivatives and certain radial differential operators that are more compatible with reproducing kernels of harmonic Bergman-Besov spaces. We show that the dual of harmonic Besov space \(b_\alpha^p\) is weighted Bloch space \(b_\beta^\infty\) under certain volume integral pairing for \(0<p<1\) and \(\alpha,\beta \in \mathbb{R}\). Our other results are about growth at the boundary and atomic decomposition.

MSC:

31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
46E15 Banach spaces of continuous, differentiable or analytic functions
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Gergün, S.; Kaptanoğlu, HT; Üreyen, AE., Reproducing kernels for harmonic Besov spaces on the ball, C R Math Acad Sci Paris, 347, 735-738 (2009) · Zbl 1179.31003
[2] Gergün, S.; Kaptanoğlu, HT; Üreyen, AE., Harmonic Besov spaces on the ball, Int J Math, 27, 9, 59 pp (2016) · Zbl 1354.31005
[3] Zhao, R.; Zhu, K., Theory of Bergman spaces in the unit ball of \(####\), Mém Soc Math Fr, 115, 103 pp (2008)
[4] Axler, S, Bourdon, P, Ramey, W.Harmonic function theory. 2nd ed.New York (NY): Springer; 2001(Graduate texts in mathematics; vol. 137). · Zbl 0959.31001
[5] Zhu, K.Spaces of holomorphic functions in the unit ball. New York (NY): Springer; 2005(Graduate texts in mathematics; vol. 226). · Zbl 1067.32005
[6] Doğan, ÖF; Üreyen, AE., Inclusion relations between harmonic Bergman-Besov and weighted Bloch spaces on the unit ball, Czech Math J, 69, 503-523 (2019) · Zbl 1513.31005
[7] Doğan, ÖF; Üreyen, AE., Weighted harmonic Bloch spaces on the ball, Complex Anal Oper Theory, 12, 5, 1143-1177 (2018) · Zbl 1395.31002
[8] Ren, G., Harmonic Bergman spaces with small exponents in the unit ball, Collect Math, 53, 83-98 (2003) · Zbl 1029.46019
[9] Coifman, RR; Rochberg, R., Representation theorems for holomorphic and harmonic functions in \(####\), Astérisque, 77, 11-66 (1980) · Zbl 0472.46040
[10] Djrbashian, AE, Shamoian, FA.Topics in the theory of \(####\) spaces. Leipzig: BSB B. G. Teubner Verlagsgesellschaft; 1998 (Teubner texts in mathematics; vol. 105).
[11] Miao, J., Reproducing kernels for harmonic Bergman spaces of the unit ball, Monatsh Math, 125, 25-35 (1998) · Zbl 0907.46020
[12] Jevtić, M.; Pavlović, M., Harmonic Bergman functions on the unit ball in \(####\), Acta Math Hungar, 85, 81-96 (1999) · Zbl 0956.32004
[13] Choe, BR; Koo, H.; Yi, H., Derivatives of harmonic Bergman and Bloch functions on the ball, J Math Anal Appl, 260, 100-123 (2001) · Zbl 0984.31004
[14] Liu, CW; Shi, CH., Invariant mean-value property and \(####\)-harmonicity in the unit ball of \(####\), Acta Math Sin, 19, 187-200 (2003) · Zbl 1031.31001
[15] Fefferman, C.; Stein, EM., \(####\) spaces of several variables, Acta Math, 129, 137-193 (1972) · Zbl 0257.46078
[16] Kuran, Ü., Subharmonic behaviour of \(####\) (p>0, h harmonic), J London Math Soc, 8, 529-538 (1974) · Zbl 0289.31007
[17] Pavlović, M., On subharmonic behaviour and oscillation of functions on balls in \(####\), Publ Inst Math, 55, 69, 18-22 (1994) · Zbl 0824.31003
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.