Bounds for maximal functions associated with rotational invariant measures in high dimensions

Journal article, 2014

ABSTRACT. In recent articles, it was proved that when mu is a finite, radial measure in Rn with a bounded, radially decreasing density, the Lp(mu) norm of the associated maximal operator 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 Lp 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 Lp bounds for every p > 2, when restricted to radially decreasing functions. On the other hand, when mu is the Gaussian measure, the Lp 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.