A good estimate for the bounded norm of little Hankel operators on weighted Bergman spaces (Q1808748)

Let \(\mathbb{D}\) be the unit disc in the complex plane, \(\alpha>-1\), and \(L^2_a(dA_\alpha)\) the Bergman space of all analytic functions in \(L^2(dA_\alpha)\) where \(dA_\alpha(z)=(\alpha+1)(1-|z|^2)^\alpha dA\), with \(dA\) the normalized Lebesgue area measure on \(\mathbb{D}\). For \(g\in L^2_a(dA_\alpha)\), define the (little) Hankel operator \(h_{z\overline g,\alpha} L^2_a(dA_\alpha)\to\overline{L^2_a(dA_\alpha)}\) with symbol \(z\overline g\) by the recipe \(h_{z\overline g,\alpha}f=P_\alpha(z\overline g f)\), where \(P_\alpha\) is the orthogonal projection of \(L^2\) onto \(\overline{L^2_a}\). Generalizing the result of \textit{F. F. Bonsall} for \(\alpha=0\) [J. Lond. Math. Soc., II. Ser. 33, 355-364 (1986; Zbl 0604.47014)], the author shows that \(h_{z\overline g,\alpha}\) extends to a bounded operator if and only if \(g\) is in the Bloch space, i.e. \(\|g\|_B:=\sup_{z\in\mathbf D}(1-|z|^2)|g'(z)|<\infty\), and, further, the norms \(\|h_{z\overline g,\alpha}\|\) and \(\|g\|_B\) are equivalent.