Hadamard Convolution and Area Integral Means in Bergman Spaces

Abstract

It is well known that if $f\in H^1$ and $g\in H^q$, where $1\le q<\infty$, then the integral means of order q of their Hadamard product $f*g$ satisfy $M_q(r,f*g)\le \Vert f\Vert _{H^1}\Vert g\Vert _{H^q}$, uniformly for each $0<r<1$, and consequently $\Vert f*g\Vert _{H^q}\le \Vert f\Vert _{H^1}\Vert g\Vert _{H^q}$. In this note, we establish similar results in Bergman spaces $A^p({\mathbb {D}})$. Namely, we show that if the fractional derivatives $D^\alpha f\in A^p({\mathbb {D}})$ and $D^{\beta } g\in A^q({\mathbb {D}})$, where $0<p\le 1$ and $p\le q<\infty$, then the area integral means of order q of $D^{\alpha +\beta -1}(f*g)$ satisfy

$$\begin{aligned} E_q\big (r,D^{\alpha +\beta -1}(f*g)\big ) \le (1-r)^{2(1-\frac{1}{p})} \Vert D^{\alpha }f\Vert _{A^p} \, \Vert D^{\beta }g\Vert _{A^q}, \quad (0<r<1). \end{aligned}$$

As an immediate consequence, we deduce that if $D^1f\in A^1({\mathbb {D}})$ and $g\in A^q({\mathbb {D}})$, where $1\le q<\infty$, then

$$\begin{aligned} \Vert f*g\Vert _{A^q}\le \Vert D^1f\Vert _{A^1}\Vert g\Vert _{A^q}. \end{aligned}$$

This last result provides an approximation theme in $A^q({\mathbb {D}})$ spaces.