Let \(B_n\) be the open unit ball in \(C^n\). For \(\alpha>-1\) let \[ dv_\alpha(z)=c_\alpha(1-|z|^2)^\alpha\,dv(z), \] where \(dv\) is the ordinary volume measure and \(c_\alpha\) is a positive normalizing constant such that \(v_\alpha(B_n)=1\). For \(0<p<\infty\) the Bergman space \(A^p_\alpha\) consists of all holomorphic functions in \(L^p(B_n,dv_\alpha)\). The norm in \(A^p_\alpha\) is denoted by \(\|f\|_{p,\alpha}\).
Let \(\mu\) be a positive Borel measure on \(B_n\). For \(\lambda>0\) and \(\alpha>-1\) we say that \(\mu\) is an \((\alpha,\alpha)\)-Carleson measure if for any two positive numbers \(p\) and \(q\) with \(q/p=\lambda\) there is a positive constant \(C\) such that \[ \int_{B_n}|f(z)|^q\,d\mu(z)\leq C\|f\|^q_{p,\alpha} \] for all \(f\in A^p_\alpha\).
The first main result of the paper shows that \(\mu\) is a \((\lambda, \alpha)\)-Carleson measure if and only if there exists a positive constant \(C\) such that \[ \int_{B_n}\prod_{i=1}^k|f_i(z)|^{q_i}\,d\mu(z)\leq C \prod_{i=1}^k\|f_i\|^{q_i}_{p_i,\alpha_i} \] for all \(f_i\in A^{p_1}_{\alpha_i}\), \(1\leq i\leq k\). Here \(k\) is any fixed positive integer, and for each \(1\leq i\leq k\), \(p_i\) and \(q_i\) are both positive and \(\alpha_i>-1\).
The second main theorem concerns generalized Toeplitz operators defined by \[ T_\mu^\beta f(z)=\int_{B_n}\frac{f(w)\,d\mu(w)}{(1-\langle z,w\rangle)^{n+1+\beta}},\qquad z\in B_n. \] Suppose \(0<p_1,p_2<\infty\), \(-1<\alpha_1,\alpha_2<\infty\), and \[ n+1+\beta>n\max\left(1,\frac1{p_1}\right)+\frac{1+\alpha_i}{p_i}, \quad i=1,2. \] It is then shown that a positive Borel measure \(\mu\) on \(B_n\) is a \((\lambda,\alpha)\)-Carleson measure if and only if \(T^\beta_\mu\) is bounded from \(A^{p_1}_{\alpha_1}\) to \(A^{p_2}_{\alpha_2}\), where \[ \lambda=1+\frac1{p_1}-\frac1{p_2},\qquad \gamma=\frac1\lambda\left(\beta+\frac{\alpha_1}{p_1}- \frac{\alpha_2}{p_2}\right). \]