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 Pérez Lázaro in Positivity 15:199–213, 2011) it was proved that when μ is a finite, radial measure in ? ⁿ with a bounded, radially decreasing density, the L ^p (μ) norm of the associated maximal operator M _μ grows to infinity with the dimension for a small range of values of p near 1. We prove that when μ 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 μ 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.