Fock projections on vector-valued $L^p$-spaces with matrix weights

In this paper, we characterize the $d\times d$ matrix weights $W$ on $\mathbb{C}^n$ such that the Fock projection $P_{\alpha}$ is bounded on the vector-valued spaces $L^p_{\alpha,W}(\mathbb{C}^n;\mathbb{C}^d)$ induced by $W$ and the Gaussian measures. It is proved that for $1\leq p\leq\infty$, the Fock projection $P_{\alpha}$ is bounded on $L^p_{\alpha,W}(\mathbb{C}^n;\mathbb{C}^d)$ if and only if $W$ satisfies a restricted $\mathcal{A}_p$-condition. Our result is new even in the scalar setting at the endpoint $p=\infty$.