Maximal inequality for high-dimensional cubes: quantitative estimates

Guillaume Aubrun (1)

1 ICJ - Institut Camille Jordan

We present lower estimates for the best constant appearing in the weak (1,1) maximal inequality in the space $(\R^n,\|\cdot\|_{\infty})$. We show that it grows to infinity faster than $(\log n)^{\kappa}$ for any $\kappa <1$. We follow the approach used by J.M.~Aldaz in a recent paper. The new part of the argument relies on Donsker's theorem identifying the Brownian bridge as the limit $(n \to \infty)$ of the empirical distribution function associated to coordinates of a point randomly chosen in the unit cube $[0,1]^n$.

Domaines: Analyse classique [math.CA]; Probabilités [math.PR]

Guillaume Aubrun. Maximal inequality for high-dimensional cubes: quantitative estimates. 2009. ⟨hal-00363618v1⟩