Let be an ergodic dynamical system on a non-atomic finite measure space. Consider the maximal function
We show that there exist and such that is not finite almost everywhere. Two consequences are derived. The bilinear Hardy-Littlewood maximal function fails to be a.e. finite for all functions . The Furstenberg averages do not converge for all pairs of functions, while by a result of J. Bourgain these averages converge for all pairs of functions with .
Cite this article
Idris Assani, Zoltán Buczolich, The bilinear Hardy-Littlewood function and Furstenberg averages. Rev. Mat. Iberoam. 26 (2010), no. 3, pp. 861–890DOI 10.4171/RMI/619