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

TL;DR

This paper investigates the behavior of maximal functions associated with rotationally invariant measures in high dimensions, revealing dimension-dependent unboundedness for certain measures and bounds for others, depending on the measure and function class.

Contribution

It establishes new dimension-dependent bounds and unboundedness results for maximal operators linked to specific rotationally invariant measures, including Lebesgue and Gaussian measures.

Findings

Maximal operator norms are unbounded in dimension for Lebesgue measure when p<2.

Dimension-free bounds exist for Lebesgue measure restricted to the unit ball for p>2.

Operator norms grow unbounded with dimension for Gaussian measure for all p>1.

Abstract

In recent articles it was proved that when $\mu$ is a finite, radial measure in $\real^n$ 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.