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We prove the uniform $\ell^2$-valued maximal inequalities for polynomialergodic averages and truncated singular operators of Cotlar type modeled overmulti-dimensional subsets of primes. In the averages case, we combine this withearlier one-parameter oscillation estimates arXiv:2212.09874 to provecorresponding multi-parameter oscillation estimates. This provides a fullerquantitative description of the pointwise convergence of the mentioned averagesand is a generalization of the polynomial Dunford-Zygmund ergodic theoremattributed to Bourgain arXiv:2209.01309. .

Vector-valued maximal inequalities and multi-parameter oscillation  inequalities for the polynomial ergodic averages along multi-dimensional  subsets of primes