Revista Matemática Iberoamericana

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

Idris Assani and Zoltán Buczolich

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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

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

First available in Project Euclid: 27 August 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37A05: Measure-preserving transformations
Secondary: 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10] 28D05: Measure-preserving transformations

Furstenberg averages bilinear Hardy-Littlewwood maximal function


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.

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