Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The $(L^{1},L^{1})$ bilinear Hardy-Littlewood function and Furstenberg
  averages   Let $(X,\mathcal{B}, \mu, T)$ be an ergodic dynamical system on a non-atomic
finite measure space. Consider the maximal function $\dis R^*:(f, g) \in
L^1\times L^1 \to R^*(f, g)(x) = \sup_{n} \frac{f(T^nx)g(T^{2n}x)}{n}.$ 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 $\frac{1}{p}+\frac{1}{q}\leq 1.$

30
The categories of this paper are: q-alg
37
The categories of this paper are: supr-con
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
31
The categories of this paper are: ao-sci
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The categories of this paper are: cmp-lg
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The categories of this paper are: mtrl-th
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The categories of this paper are: econ
4
The categories of this paper are: cond-mat
33
The categories of this paper are: plasm-ph
1
The categories of this paper are: cs
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The categories of this paper are: alg-geom
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The categories of this paper are: funct-an
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The categories of this paper are: bayes-an
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The categories of this paper are: comp-gas
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The categories of this paper are: dg-ga