Hadamard Convolution and Area Integral Means in Bergman Spaces

Abstract

It is well known that if f∈ H1 and g∈ Hq, where 1 ≤ q< ∞, then the integral means of order q of their Hadamard product f∗ g satisfy Mq(r,f∗g)≤‖f‖H1‖g‖Hq, uniformly for each 0 < r< 1 , and consequently ‖f∗g‖Hq≤‖f‖H1‖g‖Hq. In this note, we establish similar results in Bergman spaces Ap(D). Namely, we show that if the fractional derivatives Dαf∈ Ap(D) and Dβg∈ Aq(D) , where 0 < p≤ 1 and p≤ q< ∞, then the area integral means of order q of Dα + β - 1(f∗ g) satisfy Eq(r,Dα+β-1(f∗g))≤(1-r)2(1-1p)‖Dαf‖Ap‖Dβg‖Aq,(0<r<1).As an immediate consequence, we deduce that if D1f∈ A1(D) and g∈ Aq(D) , where 1 ≤ q< ∞, then ‖f∗g‖Aq≤‖D1f‖A1‖g‖Aq.This last result provides an approximation theme in Aq(D) spaces. © 2020, Springer Nature Switzerland AG.