 
Summary: Confluentes Mathematici, Vol. 1, No. 2 (2009) 169179
c World Scientific Publishing Company
MAXIMAL INEQUALITY FOR
HIGHDIMENSIONAL CUBES
GUILLAUME AUBRUN
Universit´e de Lyon, CNRS,
Institut Camille Jordan, France
aubrun@math.univlyon1.fr
Received 18 March 2009
Revised 19 August 2009
We present lower estimates for the best constant appearing in the weak (1,1) maximal
inequality in the space (Rn
, · ). We show that this constant grows to infinity faster
than (log n)1o(1) when n tends to infinity. To this end, we follow and simplify the
approach used by J. M. Aldaz. The new part of the argument relies on Donsker's theorem
identifying the Brownian bridge as the limit object describing the statistical distribution
of the coordinates of a point randomly chosen in the unit cube [0, 1]n (n large).
Keywords: Maximal inequality; highdimensional cubes; Brownian bridge.
AMS Subject Classification: 42B25
0. Introduction
