## Revista Matemática Iberoamericana

### The $(L^1,L^1)$ bilinear Hardy-Littlewood function and Furstenberg averages

#### Abstract

Let $(X,\mathcal{B}, \mu, T)$ be an ergodic dynamical system on a non-atomic finite measure space. Consider the maximal function $$R^* : (f, g) \in L^1 \times L^1 \rightarrow R^*(f, g)(x) = \sup_{n} \frac{f(T^n x) g(T^{2n} x)}{n}.$$ We show that there exist $f$ and $g$ such that $R^*(f, g)(x)$ is not finite almost everywhere. Two consequences are derived. The bilinear Hardy-Littlewood maximal function fails to be a.e. finite for all functions $(f, g)\in L^1\times L^1$. The Furstenberg averages do not converge for all pairs of $(L^1,L^1)$ functions, while by a result of J. Bourgain these averages converge for all pairs of $(L^p,L^q)$ functions with $\frac{1}{p}+\frac{1}{q} \leq 1$.

#### Article information

Source
Rev. Mat. Iberoamericana, Volume 26, Number 3 (2010), 861-890.

Dates
First available in Project Euclid: 27 August 2010

https://projecteuclid.org/euclid.rmi/1282913824

Mathematical Reviews number (MathSciNet)
MR2789368

Zentralblatt MATH identifier
1213.37006

#### Citation

Assani, Idris; Buczolich, Zoltán. The $(L^1,L^1)$ bilinear Hardy-Littlewood function and Furstenberg averages. Rev. Mat. Iberoamericana 26 (2010), no. 3, 861--890. https://projecteuclid.org/euclid.rmi/1282913824

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