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Discrete maximal functions in higher dimensions and applications to ergodic theory
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 138, Number 6, December 2016
- pp. 1495-1532
- 10.1353/ajm.2016.0045
- Article
- Additional Information
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 $1<p<\infty$ and $r>\max\{p, p/(p-1)\}$. Moreover, we obtain the estimates which are uniform in the coefficients of a polynomial mapping of fixed degree.