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
.