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

Abstract

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 antisymmetric 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 $p\in (\frac {2n}{n+1},\frac {2n}{n-1})$ .