Let \(\mathbb {D}\) be the open unit disk in the complex plane \(\mathbb {C}\) and let \(\mathcal {M}\) be a semifinite von Neumann algebra. The main result of this paper is the weak type (1, 1) inequality of the weighted Bergman projection \(P_{w}\) induced by reproducing kernels \(K_z(\zeta )=\frac{1}{(1-\bar{z} \zeta )^\gamma } \int _0^1 \frac{d \nu (r)}{1-r \bar{z} \zeta }\) , that is, if \(v\in B_{1,w}\) , then: \(\Vert P_{w}(f)\Vert _{L^{v}_{1, \infty }(\mathcal {N}_{\mathbb {D}})}\le CB_{1,w}(v)^2\Vert f\Vert _{L^{v}_1(\mathcal {N}_{\mathbb {D}})},\) where \(\mathcal {N}_{\mathbb {D}}=L_{\infty }(\mathbb {D},A_{w})\bar{\otimes } \mathcal {M}\) , \(A_{w}\) is the normalized Lebesgue area measure related to weight w. Moreover, \(P_{w}\) is bounded on \(L^{v}_{p }(\mathcal {N}_{\mathbb {D}})\) with \(1<p<\infty .\)
