Let Theta_n be the best constant in the weak (1,1) maximal inequality in R^n, with the norm ell_infinity. We show that Theta_n grows to infinity faster than (log n)^k for any k<1. The proof is an adaptation of recent work by J.M.Aldaz.
We relate some maximal function (for the counting measure on a large box of the lattice Z^n) to the statistical distribution of coordinates of a random point in [0,1]^n. The latter can be analyzed precisely since it asymptotically behaves like a Brownian bridge.
More precisely, the key estimate is the following: let x be a random point in the unit cube [0,1]^n, and consider M to be the supremum of N(r)/r^n over r>0, where N(r) is the number of integer points in the cube of edge length r centered at x (r^n is the volume of this cube). Then the probability that M is larger than any fixed number tends to 1 when n tends to infinity.
Question. I would like to know if the same is true using Euclidean balls instead of cubes (i.e. x is still a random point in [0,1]^n, but I consider the supremum of N'(r)/v(r), where N'(r) is the number of points in the Euclidean ball of size r, and v(r) the volume of this ball).
aubrun (arrobas) math. univ-lyon1. fr