A Characterization of Bergman Spaces on the Unit Ball of ℂn. II
Abstract It has been shown that a holomorphic function f in the unit ball of ℂ n belongs to the weighted Bergman space , p &gt; n + 1 + α, if and only if the function | f ( z ) – f ( w )|/|1 – 〈 z , w 〉| is in L p ( × , dv β × dv β ), where β = ( p + α – n – 1)/2 and dv β ( z ) = (1 – | z | 2 ) β dv ( z ). In this paper we consider the range 0 &lt; p &lt; n + 1 + α and show that in this case, f ∈ (i) if and only if the function | f ( z ) – f ( w )|/|1 – h z , w i| is in L p ( × , dv α × dv α ), (ii) if and only if the function | f ( z )– f ( w )|/| z – w | is in L p ( × , dv α × dv α ). We think the revealed difference in the weights for the double integrals between the cases 0 &lt; p &lt; n + 1 + α and p &gt; n + 1 + α is particularly interesting.