Kunze-Stein现象及相关极大算子的端点估计
An endpoint estimate for the Kunze-Stein phenomenon and related maximal operators
摘要：
One of the purposes of this paper is to prove that if G is a noncompact 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}). Let {K} be a maximal compact subgroup of {G} and X=G/K a symmetric space of real rank one. We will also prove that the noncentered maximal operator { \cal M}_2f(z) = \sup_{z\in B} \frac{1}{|B|} \int_{B} |f(z')|\,dz' 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 ]. The supremum in the definition of { \cal M}_2f(z) is taken over all balls containing the point z.
英文摘要：
One of the purposes of this paper is to prove that if G is a noncompact 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}). Let {K} be a maximal compact subgroup of {G} and X=G/K a symmetric space of real rank one. We will also prove that the noncentered maximal operator {\cal M}_2f(z) = \sup_{z\in B} \frac{1}{|B|} \int_{B}|f(z')|\,dz' 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]. The supremum in the definition of {\cal M}_2f(z) is taken over all balls containing the point z.
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