Abstract
We completely characterize the boundedness of area operators from the Bergman spaces \(A_\alpha ^p({\mathbb{B}_n})\) to the Lebesgue spaces \({L^q}({\mathbb{S}_n})\) for all 0 < p,q < ∞. For the case n = 1, some partial results were previously obtained by Wu in [Wu, Z.: Volterra operator, area integral and Carleson measures, Sci. China Math., 54, 2487–2500 (2011)]. Especially, in the case q < p and q < s, we obtain some characterizations for the area operators to be bounded. We solve the cases left open there and extend the results to n-complex dimension.
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The authors would like to thank the referees for making some very good suggestions.
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The first author was partially supported by NSFC (Grant Nos. 12171150, 11771139) and ZJNSF (Grant No. LY20A010008); the second author is supported by the grants MTM2017-83499-P (Ministerio de Educación y Ciencia) and 2017SGR358 (Generalitat de Catalunya); the third author was partially supported by NSFC (Grant Nos. 12171373, 12371136)
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Lv, X.F., Pau, J. & Wang, M.F. Area Operators on Bergman Spaces. Acta. Math. Sin.-English Ser. 40, 1161–1176 (2024). https://doi.org/10.1007/s10114-023-1261-4
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DOI: https://doi.org/10.1007/s10114-023-1261-4