Bounds for Maximal Functions Associated with Rotational Invariant Measures in High Dimensions

In recent articles (A. Criado in Proc. R. Soc. Edinb. Sect. A 140(3):541-552, 2010; Aldaz and Perez Lazaro in Positivity 15:199-213, 2011) it was proved that when mu is a finite, radial measure in R-n with a bounded, radially decreasing density, the L-p(mu) norm of the associated maximal operator M-mu grows to infinity with the dimension for a small range of values of p near 1. We prove that when mu is Lebesgue measure restricted to the unit ball and p < 2, the L-p operator norms of the maximal operator are unbounded in dimension, even when the action is restricted to radially decreasing functions. In spite of this, this maximal operator admits dimension-free L-p bounds for every p > 2, when restricted to radially decreasing functions. On the other hand, when mu is the Gaussian measure, the L-p operator norms of the maximal operator grow to infinity with the dimension for any finite p > 1, even in the subspace of radially decreasing functions.