Positive Toeplitz operators from a harmonic Bergman-Besov space into another. (English) Zbl 1515.47041
Summary: We define positive Toeplitz operators between harmonic Bergman-Besov spaces \(b^p_\alpha\) on the unit ball of \(\mathbb{R}^n\) for the full ranges of parameters \(0<p<\infty\), \(\alpha \in \mathbb{R}\). We give characterizations of bounded and compact Toeplitz operators taking one harmonic Bergman-Besov space into another in terms of Carleson and vanishing Carleson measures. We also give characterizations for a positive Toeplitz operator on \(b^2_{\alpha}\) to be a Schatten class operator \(S_p\) in terms of averaging functions and Berezin transforms for \(1\leq p<\infty\), \(\alpha \in \mathbb{R}\). Our results extend those known for harmonic weighted Bergman spaces.
MSC:
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
| 31B05 | Harmonic, subharmonic, superharmonic functions in higher dimensions |
| 47B10 | Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) |
Keywords:
Toeplitz operator; harmonic Bergman-Besov space; Schatten class; Carleson measure; Berezin transformReferences:
| [1] | Alpay, D.; Kaptanoğlu, HT, Toeplitz operators on Arveson and Dirichlet spaces, Integral Equ. Oper. Theory, 58, 1-33 (2007) · Zbl 1144.47022 · doi:10.1007/s00020-007-1493-1 |
| [2] | Axler, S.; Bourdon, P.; Ramey, W., Harmonic Function Theory, 2nd ed., Grad. Texts in Math. (2001), New York: Springer, New York · Zbl 0959.31001 · doi:10.1007/978-1-4757-8137-3 |
| [3] | Choe, BR; Koo, H.; Lee, Y., Positive Schatten class Toeplitz operators on the ball, Stud. Math., 189, 65-90 (2008) · Zbl 1155.47030 · doi:10.4064/sm189-1-6 |
| [4] | Choe, BR; Koo, H.; Yi, H., Positive Toeplitz operators between the harmonic Bergman spaces, Potential Anal., 17, 307-335 (2002) · Zbl 1014.47014 · doi:10.1023/A:1016356229211 |
| [5] | Choe, BR; Lee, YJ; Na, K., Toeplitz operators on harmonic Bergman spaces, Nagoya Math. J., 174, 165-186 (2004) · Zbl 1067.47039 · doi:10.1017/S0027763000008837 |
| [6] | Choe, BR; Lee, YJ; Na, K., Positive Toeplitz operators from a harmonic Bergman space into another, Tohoku Math. J. (2), 56, 2, 255-270 (2004) · Zbl 1077.47028 · doi:10.2748/tmj/1113246553 |
| [7] | Coifman, RR; Rochberg, R., Representation theorems for holomorphic and harmonic functions in \(L^p\), Astérisque, 77, 12-66 (1980) · Zbl 0472.46040 |
| [8] | Doğan, ÖF, Harmonic Besov spaces with small exponents, Complex Var. Elliptic Equ., 65, 6, 1051-1075 (2020) · Zbl 1436.31013 · doi:10.1080/17476933.2019.1652277 |
| [9] | 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 · doi:10.21136/CMJ.2018.0422-17 |
| [10] | Doğan, ÖF; Üreyen, AE, Weighted harmonic Bloch spaces on the ball, Complex Anal. Oper. Theory, 12, 5, 1143-1177 (2018) · Zbl 1395.31002 · doi:10.1007/s11785-017-0645-9 |
| [11] | Djrbashian, AE; Shamoian, FA, Topics in the Theory of \(A^p_\alpha\) Spaces, Teubner Texts in Mathematics (1988), Leipzig: BSB B. G. Teubner Verlagsgesellschaft, Leipzig · Zbl 0667.30032 |
| [12] | Fefferman, C.; Stein, EM, \(H^p\) spaces of several variables, Acta Math., 129, 137-193 (1972) · Zbl 0257.46078 · doi:10.1007/BF02392215 |
| [13] | 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 · doi:10.1016/j.crma.2009.04.016 |
| [14] | Gergün, S.; Kaptanoğlu, HT; Üreyen, AE, Harmonic Besov spaces on the ball, Int. J. Math., 27, 9, 1650070 (2016) · Zbl 1354.31005 · doi:10.1142/S0129167X16500701 |
| [15] | Jevtić, M.; Pavlović, M., Harmonic Bergman functions on the unit ball in \(\mathbb{R}^n\), Acta Math. Hung., 85, 81-96 (1999) · Zbl 0956.32004 · doi:10.1023/A:1006620929091 |
| [16] | Kuran, Ü., Subharmonic behaviour of \(|h|^p (p>0, h\) harmonic), J. Lond. Math. Soc., 8, 529-538 (1974) · Zbl 0289.31007 · doi:10.1112/jlms/s2-8.3.529 |
| [17] | Lindenstrauss, J.; Tzafriri, L., Classical Banach Spaces, Lecture Notes in Math. (1973), Berlin: Springer, Berlin · Zbl 0259.46011 |
| [18] | Liu, CW; Shi, JH, Invariant mean-value property and \(\cal{M} \)-harmonicity in the unit ball of \(\mathbb{R}^n\), Acta Math. Sin., 19, 187-200 (2003) · Zbl 1031.31001 · doi:10.1007/s10114-002-0203-9 |
| [19] | Liu, CW; Shi, JH; Ren, G., Duality for harmonic mixed-norm spaces in the unit ball of \({\mathbb{R} }^n \), Ann. Sci. Math. Que., 25, 179-197 (2001) · Zbl 1005.31003 |
| [20] | Luecking, DH, Multipliers of Bergman spaces into Lebesgue spaces, Proc. Edinb. Math. Soc. (2), 29, 125-131 (1986) · Zbl 0587.30048 · doi:10.1017/S001309150001748X |
| [21] | Luecking, DH, Embedding theorems for spaces of analytic functions via Khinchine’s inequality, Mich. Math. J., 40, 2, 333-358 (1993) · Zbl 0801.46019 · doi:10.1307/mmj/1029004756 |
| [22] | Miao, J., Reproducing kernels for harmonic Bergman spaces of the unit ball, Monatsh. Math., 125, 25-35 (1998) · Zbl 0907.46020 · doi:10.1007/BF01489456 |
| [23] | Miao, J., Toeplitz operators on harmonic Bergman spaces, Integral Equ. Oper. Theory, 27, 426-438 (1997) · Zbl 0902.47026 · doi:10.1007/BF01192123 |
| [24] | Oleinik, OL, Embedding theorems for weighted classes of harmonic and analytic functions, J. Sov. Math., 9, 228-243 (1978) · Zbl 0396.31001 · doi:10.1007/BF01578546 |
| [25] | Pau, J.; Zhao, R., Carleson measures and Toeplitz operators for weighted Bergman spaces of the unit ball, Mich. Math. J., 64, 759-796 (2015) · Zbl 1333.32007 · doi:10.1307/mmj/1447878031 |
| [26] | Ren, G., Harmonic Bergman spaces with small exponents in the unit ball, Collect. Math., 53, 83-98 (2003) · Zbl 1029.46019 |
| [27] | Pérez-Esteva, S., Duality on vector-valued weighted harmonic Bergman spaces, Stud. Math., 118, 37-47 (1996) · Zbl 0854.46022 · doi:10.4064/sm-118-1-37-47 |
| [28] | Stroethoff, K., Harmonic Bergman spaces, Holomorphic Spaces, Mathematical Sciences Research Institute Publications, 51-63 (1998), Cambridge: Cambridge University, Cambridge · Zbl 0997.46036 |
| [29] | Zhu, K., Operator Theory in Function Spaces, Second Edition, Math. Surveys and Monographs (2007), Providence: American Mathematical Society, Providence · Zbl 1123.47001 · doi:10.1090/surv/138 |
| [30] | Zygmund, A., Trigonometric Series. Vol. I, II (2002), Cambridge: Cambridge Mathematical Library, Cambridge University Press, Cambridge · Zbl 1084.42003 |
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.
