  We study the Bergman projection, $\mathcal{B}$, and the Cauchy-Szeg\H{o}
projection, $\mathcal{S}$, on bounded domains with near minimal smoothness. We
prove that $\mathcal{B}$ has the weak-type $(1,1)$ property with respect to
weighted measures assuming that the underlying domain is strongly pseudoconvex
with $C^4$ boundary and the weight satisfies the $B_1$ condition, and the same
property for $\mathcal{S}$ on domains with $C^3$ boundaries and weights
satisfying the $A_1$ condition. We also obtain weighted Kolmogorov and weighted
Zygmund inequalities for $\mathcal{B}$ and $\mathcal{S}$ in their respective
settings as corollaries.
