Convergence of ergodic-martingale paraproducts

Abstract

In this note we introduce a sequence of bilinear operators that unify ergodic averages and backward martingales in a nontrivial way. We establish its convergence in a range of $L^p$-norms and leave its a.s. convergence as an open problem. This problem shares some similarities with a well-known unresolved conjecture on a.s. convergence of double ergodic averages with respect to two commuting transformations.

arXiv comment8 pages; v2: references added and corrected following reviewer's report

References in corpus (7)

• Boundedness of the twisted paraproduct
• Some open problems on multiple ergodic averages
• Norm-variation of ergodic averages with respect to two commuting transformations
• One modification of the martingale transform and its applications to paraproducts and stochastic integrals
• Variational estimates for martingale paraproducts
• Bellman functions and $L^p$ estimates for paraproducts
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Cited by in corpus (1)

• Large dilates of hypercube graphs in the plane