Projections onto $L^p$-spaces of polyanalytic functions

Main result: for an arbitrary bounded simply connected domain $\Omega$ in $\mathbb{C}$ the subspaces $L_{n,m}^p(\Omega)$ of $L^p(\Omega)$ ($1\leqslant p<\infty$) consisting of $(m,n)$-analytic functions in $\Omega$ is complemented in $L^p(\Omega)$ (A functions $f$ on $\Omega$ is $(m,n)$-analytic if $(\partial^{m+n}/\partial\bar{z}^m\partial z^n)f=0$ in $\Omega$). It implies (due to a result of J. Lindenstrauss and A. Pelozynski) that the space $L_{n,m}^p(\Omega)$ is linearly homeomorphic to $l^p$.
In the case $m=n=1$ we get the complementedness in $L^p(\Omega)$ of the space of all harmonic $L^p$-functions in $\Omega$ — a result previously known only for smooth domains.