Weighted estimates of the Bergman projection with matrix weights

We establish a weighted inequality for the Bergman projection with matrix weights for a class of pseudoconvex domains. We extend a result of Aleman-Constantin and obtain the following estimate for the weighted norm of $P$: $$\mathrm{\parallel }P{\mathrm{\parallel }}_{L^2(\mathrm{\Omega },W)}\le C({\mathcal{B}}_2(W){)}^2\mathrm{.}$$ Here $\mathcal B_2(W)$istheBekoll\acute{e}-Bonamiconstantforthematrixweight$W$and$C$ is a constant that is independent of the weight $W$ but depends upon the dimension and the domain.