Generalized Riemann-Stieltjes operators between Hardy spaces and weighted Bergman spaces (Q2870941)

Let \(\mathbf B\) be the unit ball of the complex \(n\)-space \({\mathbb C}^n\). Given a nonnegative integer \(m\) and a holomorphic function \(g\) on \(\mathbf B\) with \(g(0)=0\), the author considers the operator \(L^{(m)}_g\) defined by \(L^{(m)}_g f(z):=\int_0^1 {\mathcal R}^m f(tz) g(tz) {dt \over t}\) for holomorphic functions \(f\) on \(\mathbf B\). Here, \({\mathcal R}=\sum_{j=1}^n z_j{{\partial}\over{\partial z_j}}\). For \(\alpha>-1\), let \(\nu_\alpha\) be the weighted measure on \(\mathbf B\) given by \(d\nu_\alpha(z):= c_\alpha (1-|z|^2)^\alpha dv(z)\), where \(\nu\) is the normalized volume measure on \(\mathbf B\) and \(c_\alpha>0\) is the normalizing constant chosen so that \(\nu_\alpha(B)=1\). For \(0<p<\infty\) and \(\alpha>-1\), let \(A^p_\alpha\) be the weighted Bergman space over \(\mathbf B\) associated with the measure \(\nu_\alpha\).NEWLINENEWLINEThe author obtains characterizations on \(g\) for the boundedness of \(L^{(m)}_g: H^2\to A^p_\alpha\), where \(H^2\) denotes the Hardy space over \(\mathbf B\). The results are restricted to the two special cases (i) \(2 \leq p<\infty\), \(m \leq {{p+\alpha+1+n}\over p} - {n \over 2}\) and (ii) \(0<p<2\), \(m< {{p+\alpha+1}\over p}\).