A generalization of multiplier theorem to the ball (Q1184910)

Let \(B=B^ n\) denote the unit ball in \(\mathbb{C}^ n\). For \(q>0\), let \(d\nu_ q(z)-{{\Gamma(n+q)} \over {\pi^ n\Gamma(q)}}(1-\| z\|^ 2)^{q-1}d\nu(z)\), where \(\nu\) is the Lebesgue measure on \(B\). The weighted Bergman space \(A_ q^ p\) is the space of all analytic functions on \(B\) that are in \(L^ p(\nu_ q)\). For \(z\in B\) and \(0<r<1\) let \(B(z,r)\) denote the pseudo-hyperbolic ball of radius \(r\) centered at \(z\). Finally, for a positive finite Borel measure \(\mu\) on \(B\) let \(k_ q(z)={{\mu(B(z,r))} \over {\nu_ q(B(z,r))}}\). In this paper the authors prove that \(A_ q^ p\subseteq L^{p'}(\mu)\), if and only if the function \(k_ q\in L^ s(\nu_ q)\), where \(1/s+p'/p=1\). For \(n=1\) and \(q=1\) this result was proved by \textit{D. Luecking} [Proc. Edinburgh Math. Soc., II. Ser. 29, 125-131 (1986; Zbl 0587.30048)]. The proof in this paper very closely follows Luecking's ideas. In fact, at the end of his article Luecking stated that the theorem remains valid in the situation considered here and that his proof would go through without change.