  We apply the Bekoll\'e-Bonami estimate for the (positive) Bergman projection
on the weighted $L^p$ spaces on the unit disk. As the consequences, we obtain
the boundedness of the Bergman projection on the weighted Sobolev space on the
symmetrized bidisk. We also improve the boundedness result of the Bergman
projection on the unweighted $L^p$ space on the symmetrized bidisk in
\cite{CKY}.
