A note on weighted Bergman spaces and the Cesaro operator (vol 159, pg 25, 2000)

Let H(D-n) be the space of holomorphic functions on the unit polydisk D-n, and let L-alpha(p,q) (D-n), where p, q > 0, alpha = (alpha(1),..., alpha(n)) with alpha(j) > -1, j = 1,..., n, be the class of all measurable functions f defined on D-n such that [GRAPHICS] where M-p(f,r) denote the p-integral means of the function f. Denote the weighted Bergman space on D-n by A(alpha)(p,q) (D-n) = L-alpha(p,q) (D-n) boolean AND H (D-n). We provide a characterization for a function f being in A(alpha)(p,q) (D-n). Using the characterization we prove the following result: Let p > 1, then the Cesaro operator is bounded on the space A(alpha)(p,p) (D-n).