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We apply the Bekollé-Bonami estimate for the (positive) Bergman projection on the weighted L p spaces on the unit disk. We then obtain the boundedness of the Bergman projection on the weighted Sobolev space on the symmetrized bidisk, by the reduction to the (positive) Bergman projection on the weighted L p space on the unit disk,. We also improve the boundedness result of the Bergman projection on the unweighted L p space on the symmetrized bidisk in [CKY].

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We observe that the recent result of implies that the canonical solution operator satisfies Sobolev estimates with a loss of n − 2 derivatives on the polydisk ∆ n and particularly is exact regular on ∆ 2 .