Banach Journal of Mathematical Analysis

Toeplitz operators on weighted harmonic Bergman spaces

Abstract

In this article, we study Toeplitz operators with nonnegative symbols on the $\mathcal{A}_{2}$-weighted harmonic Bergman space. We characterize the boundedness, compactness, and invertibility of Toeplitz operators with nonnegative symbols on this space.

Article information

Source
Banach J. Math. Anal., Volume 12, Number 4 (2018), 808-842.

Dates
Received: 1 November 2016
Accepted: 1 March 2017
First available in Project Euclid: 8 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1510110962

Digital Object Identifier
doi:10.1215/17358787-2017-0049

Mathematical Reviews number (MathSciNet)
MR3858751

Zentralblatt MATH identifier
06946293

Citation

Wang, Zipeng; Zhao, Xianfeng. Toeplitz operators on weighted harmonic Bergman spaces. Banach J. Math. Anal. 12 (2018), no. 4, 808--842. doi:10.1215/17358787-2017-0049. https://projecteuclid.org/euclid.bjma/1510110962

References

• [1] T. C. Anderson and A. Vagharshakyan, A simple proof of the sharp weighted estimate for Calderón–Zygmund operators on homogeneous spaces, J. Geom. Anal. 24 (2014), no. 3, 1276–1297.
• [2] G. R. Chacón, Toeplitz operators on weighted Bergman spaces, J. Funct. Spaces Appl. 2013, no. 753153.
• [3] B. R. Choe and Y. J. Lee, Commuting Toeplitz operators on the harmonic Bergman space, Michigan Math. J. 46 (1999), no. 1, 163–174.
• [4] E. S. Choi, Positive Toeplitz operators on pluriharmonic Bergman spaces, J. Math. Kyoto Univ. 47 (2007), no. 2, 247–267.
• [5] O. Constantin, Discretizations of integral operators and atomic decompositions in vector-valued weighted Bergman spaces, Integral Equations Operator Theory 59 (2007), no. 4, 523–554.
• [6] O. Constantin, Carleson embeddings and some classes of operators on weighted Bergman spaces, J. Math. Anal. Appl. 365 (2010), no. 2, 668–682.
• [7] R. G. Douglas, Banach Algebra Techniques in the Theory of Toeplitz Operators, CBMS Reg. Conf. Ser. Math. 15, Amer. Math. Soc., Providence, 1980.
• [8] R. G. Douglas and K. Wang, A harmonic analysis approach to essential normality of principal submodules, J. Funct. Anal. 261 (2011), no. 11, 3155–3180.
• [9] K. Guo and D. Zheng, Toeplitz algebra and Hankel algebra on the harmonic Bergman space, J. Math. Anal. Appl. 276 (2002), no. 1, 213–230.
• [10] T. P. Hytönen, The sharp weighted bound for general Calderón–Zygmund operators, Ann. of Math. (2) 175 (2012), no. 3, 1473–1506.
• [11] Ü. Kuran, Subharmonic Behaviour of $|h|^{p}$ ($p>0$, $h$ harmonic), J. Lond. Math. Soc. (2) 8 (1974), no. 3, 529–538.
• [12] D. H. Luecking, Inequalities on Bergman spaces, Illinois J. Math. 25 (1981), no. 1, 1–11.
• [13] D. H. Luecking, Equivalent norms on $L^{p}$ spaces of harmonic functions, Monatsh. Math. 96 (1983), no. 2, 133–141.
• [14] D. H. Luecking, Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives, Amer. J. Math. 107 (1985), no. 1, 85–111.
• [15] D. H. Luecking, Representation and duality in weighted spaces of analytic functions, Indiana Univ. Math. J. 34 (1985), no. 2, 319–336.
• [16] J. Miao, Toeplitz operators on harmonic Bergman spaces, Integral Equations Operator Theory 27 (1997), no. 4, 426–438.
• [17] J. Miao, Reproducing kernels for harmonic Bergman spaces of the unit ball, Monatsh. Math. 125 (1998), no. 1, 25–35.
• [18] M. Mitkovski and B. D. Wick, A reproducing kernel thesis for operators on Bergman-type function spaces, J. Funct. Anal. 267 (2014), no. 7, 2028–2055.
• [19] B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192, no. 1 (1974), 261–274.
• [20] J. Á. Peláez and J. Rättyä, Embedding theorems for Bergman spaces via harmonic analysis, Math. Ann. 362 (2015), no. 1–2, 205–239.
• [21] J. Á. Peláez and J. Rättyä, Trace class criteria for Toeplitz and composition operators on small Bergman spaces, Adv. Math. 293 (2016), no. 1, 606–643.
• [22] Y. L. Shu and X. F. Zhao, Positivity of Toeplitz operators on harmonic Bergman space, Acta Math. Sin. (Engl. Ser.) 32 (2016), no. 2, 175–186.
• [23] W. T. Sledd, A note on $L^{p}$ spaces of harmonic functions, Monatsh. Math. 106 (1988), no. 1, 65–73.
• [24] X. Zhao and D. Zheng, Invertibility of Toeplitz operators via Berezin transforms, J. Operator Theory 75 (2016), no. 2, 475–495.
• [25] K. Zhu, Operator Theory in Function Spaces, 2nd ed., Math. Surveys Monogr. 138, Amer. Math. Soc., Providence, 2007.