On weighted Bergman spaces, a uniform bound on L^m norms of S_z1 implies boundedness, and vanishing Berezin transform implies compactness.
Liu, Sharp Forelli-Rudin estimates and the norm of the Bergman projection , J
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A class of linear operators on Bergman spaces
On weighted Bergman spaces, a uniform bound on L^m norms of S_z1 implies boundedness, and vanishing Berezin transform implies compactness.