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

*(English)*Zbl 0970.43002Let \(G\) 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, 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 R. Kunze and E. Stein [Am. J. Math. 82, 1-62 (1960; Zbl 0156.37104)] in the case \(G=SL(2,{\mathbb R})\) and by M. Cowling [Ann. Math. 107, 209-234 (1978; Zbl 0371.22013)] in full generality. Assume from now on that the real rank of \(G\) is one. Using the Lorentz spaces \(L^{p,q},\) the above inclusion has been strengthened by M. Cowling, St. Meda and A. Setti (see the survey by M. Cowling in [Harmonic Analysis and Number Theory, CMS Conf. Proc. 21, 73-88 (1997; 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 \(**\)
\]
(Notice that, by interpolation, (*) is a consequence of (**).) Let \(X=G/K\) be the associated Riemannian symmetric space. Consider the non-centered maximal operator
\[
{\mathcal M}_2f(z)=\sup_{z\in B} {1\over |B|}\int_{B}|f(w)|dw,
\]
where \(f\) is a function on \(X\) and the supremum is taken over all balls \(B\) containing \(z.\) The second result of this paper is the proof that \({\mathcal M}_2\) is bounded from \(L^{2,1}(X)\) to \(L^{2,\infty}(X)\) and from \(L^p(X)\) to \(L^p(X)\) in the sharp range of exponents \(p\in (2,\infty].\)

Reviewer: Mohamed B.Bekka (Metz)