Discrete maximal functions in higher dimensions and applications to ergodic theory

We establish a higher dimensional counterpart of Bourgain’s pointwise ergodic theorem along an arbitrary integer-valued polynomial mapping. We achieve this by proving variational estimates $V_r$ on $L^p$ spaces for all $1<p<\mathrm{\infty }$ and $r>\max \{p,p\mathrm{/}(p-1)\}$. Moreover, we obtain the estimates which are uniform in the coefficients of a polynomial mapping of fixed degree.