Maximal inequality for high-dimensional cubes

Details Maximal inequality for high-dimensional cubes Maximal inequality for high-dimensional cubes

2009-02-25

arxiv.org/abs/0902.4305v2

Mathematics

Classical Analysis and ODEs

The overall presentation has been changed. To appear in Confluentes Mathematici

Scientific paper

We present lower estimates for the best constant appearing in the weak $(1,1)$ maximal inequality in the space $(\R^n,\|\cdot\|_{\iy})$. We show that this constant grows to infinity faster than $(\log n)^{1-o(1)}$ 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).

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