Hadamard convolution and area integral means in Bergman spaces (Q2175147)

Let \(f(z)=\sum_{n=0}^\infty a_n z^n\) and \(g(z)=\sum_{n=0}^\infty b_n z^n\) be holomorphic functions on the unit disc \(\mathbb{D}\subset \mathbb{C}\), and let \((f*g)(z)=\sum_{n=0}^\infty a_n b_n z^n\) be the Hadamard product of \(f\) and \(g\). A well-known norm-estimate for \(f*g\) is \(\|f*g\|_{H^q}\le \|f\|_{H^1}\|g\|_{H^q}\), \(1\le q<\infty\), where \(\|f\|_{H^p}^p=\int_{\mathbb{T}}|f(z)|^p\frac{|dz|}{2\pi}\). In this paper the authors obtain norm-estimates of Hadamard products of functions in analytic Besov spaces on \(\mathbb{D}\). For \( 0 < r \le 1 \), let \(f_r(z)=f(rz)\) and let \(E_p(r,f)=\|f_r\|_{A^p}=\left(\int_{\mathbb{D}}|f_r|^p\frac{dA}{\pi}\right)^{1/p}\). For \(\alpha\in \mathbb{R}\), let \(D^\alpha f(z)=\sum_{n=0}^\infty (n+1)^\alpha a_n z^n\) be the fractional derivative of order \(\alpha\) of \(f\). The main result in this paper states that if \(D^\alpha f\in A^p\) and \(D^\beta g\in A^q\), where \( 0< p\le 1\), \(p\le q<\infty\) and \(\alpha,\beta\in\mathbb{R}\), then \[ E_q(r, D^{\alpha+\beta-1}(f*g))\le (1-r)^{2(1-1/p)} \|D^\alpha f\|_{A^p} \|D^\beta g \|_{A^q}. \] In particular, \(\|D^{\alpha+\beta-1}(f*g)\|_{A^q} \le \|D^\alpha f\|_{A^1}\|D^\beta g\|_{A^q}\).\par As a consequence of the above result, the authors give conditions on a sequence \(\{f_m\}_m\) in order that \(\|f_m*g-g\|_{A^q}\to 0\), for all \(g\in A^q\), \(1\le q<\infty\).