{Spaces of Infinite Measure and Pointwise Convergence of the Bilinear Hilbert and Ergodic Averages Defined by $L^{p}$-Isometries

TL;DR

This paper extends the pointwise convergence results of bilinear ergodic and Hilbert averages from finite to sigma-finite measure spaces, using Lebesgue space isometries on $L^{p}$ spaces.

Contribution

It generalizes convergence theorems for bilinear averages to arbitrary measure spaces and broader $L^{p}$ settings, beyond the finite measure case.

Findings

Almost everywhere convergence of discrete bilinear ergodic averages.

Almost everywhere convergence of bilinear Hilbert averages.

Results hold for $L^{p_{1}} imes L^{p_{2}}$ with $1<p_{1},p_{2}<rac{3}{2}$.

Abstract

We generalize the respective ``double recurrence'' results of Bourgain and of the second author, which established for pairs of $L^{\infty}$ functions on a finite measure space the a.e. convergence of the discrete bilinear ergodic averages and of the discrete bilinear Hilbert averages defined by invertible measure-preserving point transformations. Our generalizations are set in the context of arbitrary sigma-finite measure spaces and take the form of a.e. convergence of such discrete averages, as well as of their continuous variable counterparts, when these averages are defined by Lebesgue space isometries and act on $L^{p_{1}}\times L^{p_{2}}$ ($ 1<p_{1},p_{2}<\infty $, $p_{1}^{-1}+p_{2}^{-1}<3/2$). In the setting of an arbitrary measure space, this yields the a.e. convergence of these discrete bilinear averages when they act on $L^{p_{1}}\times L^{p_{2}}$ and are defined by an invertible measure-preserving point transformation.