Multi-Frequency Oscillation Estimates Arising in Pointwise Ergodic Theory

Abstract

We prove essentially optimal $L^p(\mathbb{R})$-estimates for variational variants of the maximal Fourier multiplier operators considered by Bourgain in his work on pointwise convergence of polynomial ergodic averages. As a corollary of our methods, we are able to quickly extend a result of Bourgain, namely the pointwise convergence of ergodic averages of integer parts of real-variables polynomials, to a broader class of functions, previously considered in a wide range of contexts by Boshernitzan-Jones-Wierdl. Namely, the following averages converge almost everywhere \[ \frac{1}{N} \sum_{n \leq N} T^{\lfloor P(n) \rfloor} f, \; \; \; f \in L^p(X,\mu), \ P \in \mathbb{R}[\cdot], \] for any $\sigma$-finite measure space equipped with a measure-preserving transformation, $T:X \to X$, whenever $1 < p \leq \infty$ if $P$ is linear, and $4/3 < p \leq \infty$ otherwise.