Discrete harmonic analysis and applications to ergodic theory

Given $d, k\in\mathbb N$, let $P_j$ be an integer-valued polynomial of $k$ variables for every $1\le j \le d$. Suppose that $(X, \mathcal{B}, \mu)$ is a $\sigma$-finite measure space with a family of invertible commuting and measure preserving transformations $T_1, T_2,\ldots,T_{d}$ on $X$. For every $N\in\mathbb N$ and $x \in X$ we define the ergodic Radon averaging operators by setting \[ A_N f(x) = \frac{1}{N^{k}}\sum_{m \in [1, N]^k\cap\mathbb Z^k} f\big(T_1^{ P_1(m)}\circ T_2^{ P_2(m)} \circ \ldots \circ T_{d}^{ P_{d}(m)} x\big). \] We will show that for every $p > 1$ and for every function $f \in L^p(X, \mu)$, there is a function $f^*\in L^p(X, \mu)$ such that \[ \lim_{N\to\infty}A_Nf(x)=f^*(x) \] $\mu$-almost everywhere on $X$. We will achieve this by considering $r$-variational estimates. This is a joint work with Elias M. Stein and Bartosz Trojan.