Bounded Toeplitz products on the Bergman space of the unit ball in \(\mathbb C^n\) (Q2494156)

The author studies the problem of boundedness of the Toeplitz operator product \(T_fT_{\overline{g}}\) on the Bergman space \(L_a^2(B_n)\) over the unit ball \(B_n\). The main results of the paper are as follows. Theorem. Let \(f\) and \(g\) be in \(L_a^2(B_n)\). If \(T_fT_{\overline{g}}\) is bounded on \(L_a^2(B_n)\), then \[ \sup_{w \in B_n} \widetilde{| f| ^2}(w)\widetilde{| g| ^2}(w) < \infty, \] where \(\widetilde{h}\) is the Berezin transform of \(h\). Theorem. Let \(f\) and \(g\) be in \(L_a^2(B_n)\). If there is a constant \(\varepsilon >0\) such that \[ \sup_{w \in B_n} \widetilde{| f| ^{2+\varepsilon}}(w)\widetilde{| g| ^{2+\varepsilon}}(w) < \infty, \] then the operator \(T_fT_{\overline{g}}\) is bounded on \(L_a^2(B_n)\).