Tohoku Mathematical Journal
2000
March
SECOND SERIES VOL. 52, NO. 1

Tohoku Math. J.
52 (2000), 1-18

Title HARDY SPACES AND MAXIMAL OPERATORS ON REAL RANK 1 SEMISIMPLE LIE GROUPS I

Dedicated to Professor Satoru Igari on his sixtieth birthday
Author Takeshi Kawazoe

(Received May 14, 1997, revised September 9, 1999)
Abstract. Let $G$ be a real rank one connected semisimple Lie group with finite center. As well-known the radial, heat, and Poisson maximal operators satisfy the $L^p$-norm inequalities for any $p>1$ and a weak type $L^1$ estimate. The aim of this paper is to find a subspace of $L^1(G)$ from which they are bounded into $L^1(G)$. As an analogue of the atomic Hardy space on the real line, we introduce an atomic Hardy space on $G$ and prove that these maximal operators with suitable modifications are bounded from the atomic Hardy space on $G$ to $L^1(G)$.

1991 Mathematics Subject Classification. Primary 22E30; Secondary 42C20.

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