M. Hormozi

On $L^p$ norms of maximal operators

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 $(T_t)_{t\ge0}$ on a nonatomic $\sigma$-finite measure space $(X,\mathcal F,\mu)$ has maximal $L^p...

Endpoint Failure for the Adjoint Vertical Riesz Transform on ax+b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ax+b$

Dimension-free estimates for full discrete maximal functions associated with Euclidean balls and spheres

We prove that the full discrete Hardy-Littlewood maximal operator associated with Euclidean balls satisfies dimension-free bounds on $\ell^p(\mathbb Z^d)$ for every $1<p<\infty$. We also establish analogous dimension-free bounds for the full discrete spherical maximal operator when $d\geq 5$ and $2\leq p<\infty$. The m...