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An endpoint estimate for the Kunze-Stein phenomenon and related maximal operators. (English) Zbl 0970.43002
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, 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 $$p\in (2,\infty].$$

MSC:
 43A85 Harmonic analysis on homogeneous spaces 22E46 Semisimple Lie groups and their representations
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