  We prove the uniform $\ell^2$-valued maximal inequalities for polynomial
ergodic averages and truncated singular operators of Cotlar type modeled over
multi-dimensional subsets of primes. In the averages case, we combine this with
earlier one-parameter oscillation estimates arXiv:2212.09874 to prove
corresponding multi-parameter oscillation estimates. This provides a fuller
quantitative description of the pointwise convergence of the mentioned averages
and is a generalization of the polynomial Dunford-Zygmund ergodic theorem
attributed to Bourgain arXiv:2209.01309. .
