Area Operators on Bergman Spaces

We completely characterize the boundedness of area operators from the Bergman spaces A_αP(Bn) to the Lebesgue spaces Lq(Sn) for all 0 <p,q <∞.For the case n=1,some partial results were previously obtained by Wu in [Wu,Z.:Volt err a 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.