An endpoint estimate for the Kunze-Stein phenomenon and related maximal operators

Alexandru D. Ionescu

Review from Zentralblatt MATH:

Let $G$ be a semisimple Lie group with finite centre. A central result in the theory of convolution operators on such a group is the Kunze-Stein phenomenon which says that, if $p\in [1,2),$ then $$ L^2(G)\ast L^p(G)\subseteq L^2(G). $$ (Equivalently, {\it all} matrix coefficients of the regular representation of $G$ belong to $L^{2+\varepsilon}(G)$ for all $\varepsilon>0.$) This is a feature of the world of semisimple groups and fails for, say, all amenable non-compact groups. It was proved by {\it R. Kunze} and {\it E. Stein} [Am. J. Math. 82, 1-62 (1960; Zbl 0964.22008)]) as follows: $$ L^{p,u}(G)\ast L^{p,v}(G)\subseteq L^{p,w}(G),\tag $*$ $$ where $p\in (1,2), 1\leq u,v, w\leq \infty$ and $1+1/w\leq 1/u+1/v.$ The first result in the paper under review is an endpoint estimate for the above inclusion showing that for $p=2$ one has $$ L^{2,1}(G)\ast L^{2,1}(G)\subseteq L^{2,\infty}(G). \tag $

Reviewed by Mohamed B.Bekka

Keywords: Kunze-Stein phenomenon; convolution operator; semisimple group; maximal operator

Classification (MSC2000): 43A85 22E46

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