Variational Estimates for Bilinear Ergodic Averages Along Sublinear Sequences

TL;DR

This paper establishes sharp variational bounds for bilinear ergodic averages along sublinear sequences, extending the understanding of convergence and maximal inequalities in ergodic theory.

Contribution

It provides the first comprehensive variational estimates for bilinear ergodic averages along sublinear sequences, covering the full expected range of exponents and variation parameters.

Findings

Proves long variational estimates for bilinear ergodic averages with optimal range.

Establishes bilinear maximal inequalities in L^p spaces.

Identifies sharp bounds up to endpoint cases.

Abstract

We prove long variational estimates for the bilinear ergodic averages \[ A_{N;X}(f,g)(x) = \frac{1}{N} \sum_{n=1}^N f(T^{\lfloor \sqrt{n} \rfloor}x) g(T^nx) \] on an arbitrary measure preserving system $(X,\mu,T)$ for the full expected range, i.e. whenever $f \in L^{p_1}(X)$ and $g \in L^{p_2}(X)$ with $1<p_1,p_2<\infty$. In particular, if $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$ we show that the long $r$-variation of $A_{N;X}$ maps $L^{p_1}(X) \times L^{p_2}(X)$ into $L^p(X)$ for any $p>\frac{1}{2}$, which is sharp up to the endpoint. If $p \geq 1$ we obtain long variational estimates for the full expected range $r>2$ and if $p<1$ we obtain a range of $r>2+\varepsilon_{p_1,p_2}$ where $\varepsilon_{p_1,p_2}>0$ depends only on $p_1$ and $p_2$. As a consequence, we obtain bilinear maximal estimates \[ \left\| \sup_{N \in \mathbb{N}} |A_{N;X}(f,g)| \right\|_{L^p(X)} \leq C_{p_1,p_2} \|f\|_{L^{p_1}(X)} \|g\|_{L^{p_2}(X)} \] for any $1<p_1,p_2 \leq \infty$.