JournalsrmiVol. 26, No. 3pp. 861–890

The (L1,L1)(L^1,L^1) bilinear Hardy-Littlewood function and Furstenberg averages

  • Idris Assani

    University of North Carolina at Chapel Hill, USA
  • Zoltán Buczolich

    Eötvös Loránd University, Budapest, Hungary
The $(L^1,L^1)$ bilinear Hardy-Littlewood function and Furstenberg averages cover
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Abstract

Let (X,B,μ,T)(X,\mathcal{B}, \mu, T) be an ergodic dynamical system on a non-atomic finite measure space. Consider the maximal function

R:(f,g)L1×L1R(f,g)(x)=supnf(Tnx)g(T2nx)n.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 ff and gg such that R(f,g)(x)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)L1×L1(f, g)\in L^1\times L^1. The Furstenberg averages do not converge for all pairs of (L1,L1)(L^1,L^1) functions, while by a result of J. Bourgain these averages converge for all pairs of (Lp,Lq)(L^p,L^q) functions with 1p+1q1\frac{1}{p}+\frac{1}{q} \leq 1.

Cite this article

Idris Assani, Zoltán Buczolich, The (L1,L1)(L^1,L^1) bilinear Hardy-Littlewood function and Furstenberg averages. Rev. Mat. Iberoam. 26 (2010), no. 3, pp. 861–890

DOI 10.4171/RMI/619