Multi-Frequency Oscillation Estimates Arising in Pointwise Ergodic Theory

Abstract

Abstract We prove essentially optimal $$L^p(\mathbb {R})$$ L p ( 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 $$\begin{aligned} \frac{1}{N} \sum _{n \le N} T^{\lfloor P(n) \rfloor } f, \; \; \; f \in L^p(X,\mu ), \ P \in \mathbb {R}[\cdot ], \end{aligned}$$ 1 N ∑ n ≤ N T ⌊ P ( n ) ⌋ f , f ∈ L p ( X , μ ) , P ∈ R [ · ] , for any $$\sigma $$ σ -finite measure space equipped with a measure-preserving transformation, $$T:X \rightarrow X$$ T : X → X , whenever $$1 < p \le \infty $$ 1 < p ≤ ∞ if P is linear, and $$4/3 < p \le \infty $$ 4 / 3 < p ≤ ∞ otherwise.