Area Operators on Bergman Spaces

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.