Recently, Zhang et al. generalized the Forelli-Rudin type operator \(\begin{aligned} S_{\lambda ,\tau ,c}f(\xi )=(1-|\xi |^{2})^{\lambda }\int _{B_{n}}\frac{(1-|u|^{2})^{\tau }\ f(u)}{|1-\langle \xi ,u\rangle |^{c}}\ dv(u) \end{aligned}\) to \(\displaystyle {S_{\lambda ,\tau ,c,k,k'}f(\xi )=\int _{B_{n}}K(\xi ,u)f(u)\ dv(u) \ \ (\xi \in B_{n})}\) , the kernel \(\begin{aligned} K(\xi ,u)=\frac{(1-|\xi |^{2})^{\lambda }(1-|u|^{2})^{\tau }}{ |1-\langle \xi ,u\rangle |^{c}}\left| \log \frac{e}{1-\langle \xi ,\varphi _{\xi }(u)\rangle }\right| ^{k} \log ^{k'}\frac{e}{1-|u|^{2}} \ \ (\xi ,u\in B_{n}). \end{aligned}\) They discussed the conditions for which \(S_{\lambda ,\tau ,c,k,k'}\) is bounded from \(L^{p}(B_{n}, dv_{t})\) to \(L^{q}(B_{n}, dv_{t})\) for \(1\le p\le q\le +\infty \) and \(t>-1\) . However, the case \(1\le q<p\le +\infty \) was not considered. In this paper, we will discuss this problem, and completely characterize the boundedness of \(S_{\lambda ,\tau ,c,k,k'}\) from \(L^{p}(B_{n}, dv_{\alpha })\) to \(L^{q}(B_{n}, dv_{\beta })\) for \(1\le q< p\le +\infty \) , where \(\lambda , \tau , c, k, k', \alpha , \beta \) are real numbers. The results partially generalize some previous results on Forelli-Rudin type operators.
