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Zbl 0899.43004
Blank, Brian E.; Fan, Dashan
Hardy spaces on compact Lie groups. (English)
[J] Ann. Fac. Sci. Toulouse, VI. Sér., Math. 6, No.3, 429-479 (1997). ISSN 0240-2963

The authors study the Hardy space $H^p(G)$ of a connected semisimple compact Lie group $G$. They first characterize this space for $0< p\le 1$ by maximal functions based on Poisson kernel, using elaborate estimations of kernels. Next, they treat the atomic characterization of $H^p(G)$. They define $p$-atoms $(0< p\le 1)$ in $L^q(G)$ $(1\le q\le\infty)$ by support, size and cancellation properties, and prove that any distribution in $H^p(U(n))$ has an atomic decomposition. The existence of a faithful finite-dimensional unitary representation then allows them to transfer the result to $G$.\par They also characterize $H^p(G)$ by maximal functions based on heat kernel for $0< p\le 1$, by the maximal Bochner-Riesz operator $S^\delta_*$ for $\delta> n/p-(n+ 1)/2$ $(n= \dim_{\bbfR}G)$ and, for $p= 1$, by certain nonsmooth kernels satisfying a Dini condition.\par Throughout the paper, the analysis previously undertaken by {\it M. Cowling}, {\it A. M. Mantero} and {\it F. Ricci} [Rend. Circ. Mat. Palermo, II. Ser. 31, 145-158 (1982; Zbl 0492.43006)] and by {\it J. L. Clerc} [Lect. Notes Math. 1243, 86-107 (1987; Zbl 0625.43003)] turns out to be influential.
[H.Fujiwara (Iizuka)]

MSC 2000:: *43A77 Analysis on general compact groups
42B30 Hp-spaces (Fourier analysis)

Keywords: Hardy space; semisimple compact Lie group; maximal functions; Poisson kernel; unitary representation; heat kernel; maximal Bochner-Riesz operator; Dini condition

Citations: Zbl 0492.43006; Zbl 0625.43003

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