On Lp norms of maximal operators
Abstract
We study the exact Lp 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 (Tt)t≥0 on a nonatomic σ-finite measure space (X,ℱ,μ) has maximal Lp norm p'=p/(p−1) for every 1<p<∞ whenever f↦ Tt0(1Ef) is compact on L2(μ) for some t0>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 ℝ. In particular, the heat and Poisson maximal operators on every complete connected Riemannian manifold of positive dimension without boundary have exact Lp norm p', with no curvature or stochastic completeness assumptions. Combining our approach with previous work of the first and last authors we also prove that p' is the limit, as d→ ∞, of the Lp norms B(p,d) of the centered Hardy-Littlewood maximal operators on ℝd. On the other hand, we show that the norm of this maximal operator is strictly larger than p', in dimensions d≥2.
Related
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BibTeX
@misc{ciccone2026norms,
title = {{On $L^p$ norms of maximal operators}},
author = {Valentina Ciccone and Mahdi Hormozi and José Madrid and Jakub Niksiński and Błażej Wróbel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10247},
eprint = {2610.10247},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10247},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On L^p norms of maximal operators
AU - Valentina Ciccone
AU - Mahdi Hormozi
AU - José Madrid
AU - Jakub Niksiński
AU - Błażej Wróbel
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10247
UR - https://arxiv.org/abs/2610.10247
AB - 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≥0 on a nonatomic σ-finite measure space (X,ℱ,μ) has maximal L^p norm p'=p/(p−1) for every 1<p<∞ whenever f↦ T_t_0(1_Ef) is compact on L^2(μ) 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 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', with no curvature or stochastic completeness assumptions. Combining our approach with previous work of the first and last authors we also prove that p' is the limit, as d→ ∞, of the L^p norms B(p,d) of the centered Hardy-Littlewood maximal operators on ℝ^d. On the other hand, we show that the norm of this maximal operator is strictly larger than p', in dimensions d≥2.
ER -