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If a bounded domain can be covered by the polydisk through a rational properholomorphic map, then the Bergman projection is $L^p$-bounded for $p$ in acertain range depending on the ramified rational covering. This result can beapplied to the symmetrized polydisk and to the Hartogs triangle with exponent$\gamma$.

$L^p$ regularity of the Bergman Projection on domains covered by the polydisk