Revista Matemática Iberoamericana

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Volume 26, Issue 3, 2010, pp. 861–890
DOI: 10.4171/RMI/619

Published online: 2010-12-31

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

Idris Assani[1] and Zoltán Buczolich[2]

(1) University of North Carolina at Chapel Hill, USA
(2) Eötvös Loránd University, Budapest, Hungary

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 $\frac{1}{p}+\frac{1}{q} \leq 1$.

Keywords: Furstenberg averages, bilinear Hardy–Littlewood maximal function

Assani Idris, Buczolich Zoltán: The $(L^1,L^1)$ bilinear Hardy-Littlewood function and Furstenberg averages. Rev. Mat. Iberoam. 26 (2010), 861-890. doi: 10.4171/RMI/619