Let \(t=(t_1, \dots, t_n)\) be a real vector, where every \(t_j\) satisfies \(t_j>-1\) and consider the measure on \(\mathbb{D}^n\) given by \(dv_t(z)=c_t \prod_{j=1}^n (1-| z_j| ^2)^{t_j}\,dv(z_j),\) where \(c_t=\prod_{j=1}^n(t_j+1)\) and \(dv\) is the Lebesgue area measure on \(\mathbb{D},\) normalized so that the measure of \(\mathbb{D}\) is \(1.\) Let \(A^2_t\) denote the Bergman space of \(L^2\) holomorphic functions on \(\mathbb{D}^n\) with respect to the measure \(dv_t.\) The Berezin transform is the operator defined on \(L^1(\mathbb{D}^n, dv_t)\) by \(B_t(f)(w)=\int_{\mathbb{D}^n}f(z)| k^t_w(z)| ^2\, dv_t(z),\) where \(k^t_w\) is the normalized reproducing kernel of \(A^2_t.\) The main result of this paper is a generalization of results of K. Stroethoff and A. D. Zheng [J. Math. Anal. Appl. 278, No. 1, 125–135 (2003; Zbl 1051.47025)] and reads as follows:
(a) If \(f,g \in A^2_t\) and the composition of the Toeplitz operators \(T_f T_{\overline g}\) is bounded on \( A^2_t,\) then \[ \sup_{w\in \mathbb{D}^n} B_t(| f| ^2)(w) B_t(| g| ^2)(w)< \infty. \]
(b) If \(f,g \in A^2_t\) and there exists \(\epsilon >0\) such that \[ B_t(| f| ^{2+\epsilon})(w) B_t(| g| ^{2+\epsilon})(w)< \infty \] then \( T_f T_{\overline g}\) is bounded.