Abstract
Suppose that \(\omega \) is a radial weight on the unit disk that satisfies both forward and reverse doubling conditions. Using Carleson measures and T1-type conditions, we obtain necessary and sufficient conditions of the positive Borel measure \(\mu \) such that the Toeplitz operator \(T_{\mu ,\omega }:L^p_a(\omega )\rightarrow L_a^1(\omega )\) is bounded and compact for \(0<p\le 1\). In addition, we obtain a bump condition for the bounded Toeplitz operators with \(L^1(\omega )\) symbol on \(L^1_a(\omega )\). This generalizes a result of Zhu in (J Funct Anal 87(1):31-50, 1989).
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Acknowledgements
We thank Professor D. H. Luecking for his explanation of some details in [25]. This work is partially supported by National Natural Science Foundation of China. Y. Duan thanks School of Mathematics of Fudan University for the support of his visit to Shanghai. Z. Wang thanks School of Mathematics and Statistics, Northeast Normal University for the support of his visit to Changchun. We thank the anonymous referees for many valuable suggestions which make the paper more readable.
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Duan, Y., Guo, K., Wang, S. et al. Toeplitz Operators on Weighted Bergman Spaces Induced by a Class of Radial Weights. J Geom Anal 32, 39 (2022). https://doi.org/10.1007/s12220-021-00777-z
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DOI: https://doi.org/10.1007/s12220-021-00777-z