$L^p$ regularity of the Bergman projection on the symmetrized polydisc

We study the $L^p$ regularity of the Bergman projection $P$ over the symmetrized polydisc in ${\mathbb{C}}^n$. We give a decomposition of the Bergman projection on the polydisc and obtain an operator equivalent to the Bergman projection over anti-symmetric function spaces. Using it, we obtain the $L^p$ irregularity of $P$ for $p=\frac{2n}{n-1}$ which also implies that $P$ is $L^p$ bounded if and only if $p\in (\frac{2n}{n+1},\frac{2n}{n-1})$.