Hadamard products in weighted Bergman spaces

Abstract

Using the fractional derivatives Dt of Hardy-Littlewood and auxiliary functions ξγ,r and ηγ,r, we study the growth of the Hadamard product f⁎g in the weighted Bergman space Aγq(D). We precisely evaluate the growth of dilations of Dα+β(f⁎g⁎ξγ,r), whenever Dαf∈Aγp(D) and Dβg∈Aγq(D). The main result has numerous special cases which are interesting in their own right. In particular, we show that ‖f⁎g‖Aγq(D)≤‖f⁎ηγ,1‖Aγ1(D)‖g‖Aγq(D).