On $L^p$ norms of maximal operators

Abstract

We study the exact $L^p$ norms for a number of maximal operators, which are prominent in harmonic analysis, semigroup theory, and probability. We prove that a strongly continuous positive symmetric sub-Markovian semigroup $(T_t{)}_{t\ge 0}$ on a nonatomic $\sigma$-finite measure space $(X,\mathcal{F},\mu )$ has maximal $L^p$ norm $p^{\prime }=p/(p-1)$ for every $1<p<\infty$ whenever $f\mapsto T_{t_0}(1_{E}f)$ is compact on $L^2(\mu )$ for some $t_0>0$ and some measurable set $E$ of positive measure. The lower bound follows by approximating finite dyadic conditional expectations at suitable semigroup times; the upper bound is Stein's maximal inequality in its sub-Markovian form. Important examples of such maximal operators we discuss include maximal functions of various semigroups (heat, Poisson, Ornstein--Uhlenbeck, Schrödinger, Dunkl, Laguerre, Jacobi) as well as the centered Hardy-Littlewood maximal operator on $\mathbb{R}.$ In particular, the heat and Poisson maximal operators on every complete connected Riemannian manifold of positive dimension without boundary have exact $L^p$ norm $p^{\prime }$, with no curvature or stochastic completeness assumptions. Combining our approach with previous work of the first and last authors we also prove that $p^{\prime }$ is the limit, as $d\to \infty ,$ of the $L^p$ norms $B(p,d)$ of the centered Hardy-Littlewood maximal operators on ${\mathbb{R}}^d.$ On the other hand, we show that the norm of this maximal operator is strictly larger than $p^{\prime },$ in dimensions $d\ge 2.$

https://arxiv.org/abs/2609.12440 was merged into this article, 22 pages