Quantitative norm convergence of double ergodic averages associated with two commuting group actions

Abstract

We study double averages along orbits for measure preserving actions of ${\mathbb{A}}^{\omega }$, the direct sum of countably many copies of a finite abelian group $\mathbb{A}$. In this article we show an $L^p$ norm-variation estimate for these averages, which in particular reproves their convergence in $L^p$ for any finite $p$ and for any choice of two $L^{\infty }$ functions. The result is motivated by recent questions on quantifying convergence of multiple ergodic averages.

15 pages; v2: referee comments incorporated, small typos corrected, research support acknowledged

References in corpus (3)

• Bellman function technique for multilinear estimates and an application to generalized paraproducts
• Norm convergence of multiple ergodic averages on amenable groups
• Multiple recurrence and convergence results associated to ${\mathbb{F}}_p^{\omega }$-actions

Cited by in corpus (5)

• Norm-variation of ergodic averages with respect to two commuting transformations
• Norm-variation of cubic ergodic averages
• T(1) theorem for dyadic singular integral forms associated with hypergraphs
• Large dilates of hypercube graphs in the plane
• Norm-variation of triple ergodic averages for commuting transformations