The commutants of certain Toeplitz operators on weighted Bergman spaces

For $\alpha >-1$, let $A_{\alpha }^2$ be the corresponding weighted Bergman space of the unit ball in ${\mathbb{C}}^n$. For a bounded measurable function $f$, let $T_f$ be the Toeplitz operator with symbol $f$ on $A_{\alpha }^2$. This paper describes all the functions $f$ for which $T_f$ commutes with a given $T_g$, where $g(z)=z_1^{L_1}\mathrm{.}\mathrm{.}\mathrm{.}{z}_n^{L_n}$ for strictly positive integers $L_1,\mathrm{.}\mathrm{.}\mathrm{.},L_n$, or $g(z)=\mathrm{\mid }{z}_1{\mathrm{\mid }}^{s_1}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{\mid }{z}_n{\mathrm{\mid }}^{s_n}h(\mathrm{\mid }z\mathrm{\mid })$ for non-negative real numbers $s_1,\mathrm{.}\mathrm{.}\mathrm{.},s_n$ and a bounded measurable function $h$ on $[0,1)$.