科研之友
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摘要
It has been shown that a holomorphic function f in the unit ball B-n of C-n belongs to the weighted Bergman space A(alpha)(p), p %26gt; n + 1 + alpha, if and only if the function vertical bar f(z) - f(w)vertical bar/vertical bar 1 - %26lt; z, w %26gt;vertical bar is in L-p(B-n x B-n, dv(beta) x dv(beta)), where beta = (p + alpha - n - 1)/2 and dv(beta)(z) = (1 - vertical bar z vertical bar(2))(beta) dv(z). In this paper we consider the range 0 %26lt; p %26lt; n + 1 + alpha and show that in this case, f is an element of A(alpha)(p) (i) if and only if the function vertical bar f(z) - f(w)vertical bar/vertical bar 1 - %26lt; z, w %26gt;vertical bar is in L-p(B-n x B-n, dv(alpha) x dv(alpha)), (ii) if and only if the function vertical bar f(z) - f(w)vertical bar/vertical bar z - w vertical bar is in L-p(B-n x B-n, dv(alpha) x dv(alpha)). We think the revealed difference in the weights for the double integrals between the cases 0 %26lt; p %26lt; n + 1 + alpha and p %26gt; n + 1 + alpha is particularly interesting.
