Fock projections on vector-valued Lᵖ-spaces with matrix weights

In this paper, we characterize the d× d matrix weights W on Cⁿ such that the Fock projection Pα is bounded on the vector-valued spaces Lᵖα,W(Cⁿ;Cᵈ) induced by W. It is proved that for 1≤ p<∞, the Fock projection Pα is bounded on Lᵖα,W(Cⁿ;Cᵈ) if and only if W satisfies a restricted Aₚ-condition. In particular, when $p=1$, our result establishes a strong (1,1) type estimate for the Fock projections, which is quite different with the case of Calder\'{o}n--Zygmund operators.