In this paper, it is shown that the harmonic Bergman projection $P^h_{\omega}$, induced by a radial weight $\omega,$ is bounded and onto from $L^{\infty}(\mathbb{D})$ to the harmonic Bloch space $\mathcal{B}_{h}$ if and only if $\omega\in \mathcal{D}$, which is a class of radial weights satisfying the two-sided doubling conditions. As an application, the bounded and compact positive Toeplitz operators $T^h_{\mu,\omega}$ on the endpoint case weighted harmonic Bergman space $L^1_{h,\omega}(\mathbb{D})$ are characterized.
Yongjiang DUAN
,
Sawlet JUNIS
,
Na ZHAN
. BERGMAN PROJECTION AND TOEPLITZ OPERATORS ON WEIGHTED HARMONIC BERGMAN SPACES INDUCED BY DOUBLING WEIGHTS[J]. Acta mathematica scientia, Series B, 2025
, 45(4)
: 1497
-1513
.
DOI: 10.1007/s10473-025-0414-1
[1] Agbor D, Békollé D, Tchoundja E. Bounded and compact operators on the Bergman space {$L^1_a$} in the unit disk of $\Bbb C$. Afr Diaspora J Math, 2011, 11: 1-29
[2] Bekollé D, Bonami A. Inégalités à poids pour le noyau de Bergman. C R Acad Sci Paris Sér A-B, 1978, 286: 775-778
[3] Bonet J, Lusky W, Taskinen J. Unbounded Bergman projections on weighted spaces with respect to exponential weights. Integral Equations Operator Theory, 2021, 93: Art 61
[4] Choe B R, Lee Y J, Na K. Toeplitz operators on harmonic Bergman spaces. Nagoya Math J, 2004, 174: 165-186
[5] Colonna F. The Bloch constant of bounded harmonic mappings. Indiana Univ Math J, 1989, 38: 829-840
[6] Dostanić M. Unboundedness of the Bergman projections on $L^p$ spaces with exponential weights. Proc Edinb Math Soc, 2004, 47(1): 111-117
[7] Duan Y J, Guo K Y, Wang S Y, Wang Z P. Toeplitz operators on weighted Bergman spaces induced by a class of radial weights. J Geom Anal, 2022, 32: Art 39
[8] Duan Y J, Guo K Y, Wang S Y, Wang Z P. Toeplitz operators on a class of radially weighted harmonic Bergman spaces. Potential Anal, 2023, 59: 1621-1641
[9] Duan Y J, Rättyä J, Wang S Y, Wu F L. Two weight inequality for Hankel form on weighted Bergman spaces induced by doubling weights. Adv Math, 2023, 431: Art 109249
[10] Luecking D H. Trace ideal criteria for Toeplitz operators. J Funct Anal, 1987, 73: 345-368
[11] Luecking D H. Embedding theorems for spaces of analytic functions via Khinchine's inequality. Michigan Math J, 1993, 40: 333-358
[12] Lv X F, Arroussi H. Toeplitz operators on Bergman spaces with exponential weights for $0<p\leq1$. Bull Sci Math, 2021, 173: Art 103068
[13] Martín Reyes F J, Ortega P, Peláez J, Rättyä J. One weight inequality for Bergman projection and Calderón operator induced by radial weight. Proc Amer Math Soc, 2023, 151: 5189-5203
[14] Miao J. Toeplitz operators on harmonic Bergman spaces. Integral Equations Operator Theory, 1997, 27: 426-438
[15] Pavlović M.Function Classes on the Unit Disc. An Introduction. Berlin: De Gruyter, 2014
[16] Peláez J. Small weighted Bergman spaces. arXiv.1507.07182,2015
[17] Peláez J, Rättyä J.Weighted Bergman Spaces Induced by Rapidly Increasing Weights. Providence, RI: Mem Amer Math Soc, 2014
[18] Peláez J, Rättyä J. Two weight inequality for Bergman projection. J Math Pures Appl, 2016, 105: 102-130
[19] Peláez J, Rättyä J. Bergman projection induced by radial weight. Adv Math, 2021, 391: Art 107950
[20] Peláez J, Rättyä J, Sierra K. Berezin transform and Toeplitz operators on Bergman spaces induced by regular weights. J Geom Anal, 2018, 28: 656-687
[21] Wang X L, Liu T S. Toeplitz operators on Bloch-type spaces in the unit ball of ${C}^n$. J Math Anal Appl, 2010, 368: 727-735
[22] Wang Z P, Zhao X F. Toeplitz operators on weighted harmonic Bergman spaces. Banach J Math Anal, 2018, 12: 808-842
[23] Zhu K H.Operator Theory in Function Spaces. Providence RI: American Mathematical Society, 2007