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

We prove the uniform ${\mathrm{\ell }}^2$-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. .