Let
G be a real rank one connected semisimple Lie group with finite center. We introduce a real Hardy space
H1 (
G//
K) on
G as the space consisting of all
K-bi-invariant functions
f on
G whose radial maximal functions
Mφf are integrable on
G. We shall obtain a relation between
H1 (
G//
K) and
H1(
R), the real Hardy space on the real line
R, via the Abel transform on
G and give a characterization of
H1 (
G//
K).
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