  Our first result is a noncommutative form of Jessen/Marcinkiewicz/Zygmund
theorem for the maximal limit of multiparametric martingales or ergodic means.
It implies bilateral almost uniform convergence with initial data in the
expected Orlicz spaces. A key ingredient is the introduction of the $L_p$-norm
of the $\limsup$ of a sequence of operators as a localized version of a
$\ell_\infty/c_0$-valued $L_p$-space. In particular, our main result gives a
strong $L_1$-estimate for the $\limsup$, as opposed to the usual weak
$L_{1,\infty}$-estimate for the $\sup$.
  Let $\mathcal{L} \mathbf{F}_2$ denote the free group algebra and consider the
free Poisson semigroup generated by the usual length function. It is an open
problem to determine the largest class inside $L_1(\mathcal{L} \mathbf{F}_2)$
for which this semigroup converges to the initial data. Currently, the best
known result is $L \log^2 L(\mathcal{L} \mathbf{F}_2)$. We improve this by
adding to it the operators in $L_1(\mathcal{L} \mathbf{F}_2)$ spanned by words
without signs changes. Contrary to other related results in the literature,
this set has exponential growth. The proof relies on our estimates for the
noncommutative $\limsup$ together with new transference techniques.
  We also establish a noncommutative form of C\'ordoba/Feffermann/Guzm\'an
inequality for the strong maximal. More precisely, a weak $(\Phi,\Phi)$
inequality for noncommutative multiparametric martingales and $\Phi(s) = s (1 +
\log_+ s)^{2 + \varepsilon}$. This logarithmic power is an
$\varepsilon$-perturbation of the expected optimal one. The proof combines a
refinement of Cuculescu's construction with a quantum probabilistic
interpretation of de Guzm\'an's argument. The commutative form of our argument
gives the simplest known proof of this classical inequality. A few interesting
consequences are derived for Cuculescu's projections.
