Topic: An Endpoint Estimate for the Kunze-Stein Phenomenon May 24

and Related Maximal Operators

Presenter: Alexander Ionescu, Princeton University

Abstract: One of the purposes of this talk is to prove that if $G$ is a non-compact connected semisimple Lie group of

real rank one with finite center then $L^{2,1}(G)\ast L^{2,1}(G)\subseteq L^{2,\infty}(G)$. This is an endpoint estimate

for the Kunze-Stein phenomenon on the group $G$. Under the same assumptions on the group $G$, we will also prove

that the noncentered maximal operator

$$\mathcal{M}_2f(z)=\sup_{z\in B}\frac{1}{|B|}\int_{B}f(z')dz'$$

is bounded from $L^{2,1}(G/K)$ to $L^{2,\infty}(G/K)$ and from $L^p(G/K)$ to $L^p(G/K)$ in the sharp range of

exponents $p\in(2,\infty]$. Here $K$ is a maximal compact subgroup of $G$ and the supremum in the definition of

$\mathcal{M}_2f(z)$, is taken over all balls containing the point $z$.