We study the operator
\[
H
f
(
x
)
=
2
−
x
∫
0
+
∞
2
y
f
(
y
)
x
−
y
d
y
\mathcal {H}f(x) = {2^{ - x}}\int _0^{ + \infty } {\frac {{{2^y}f(y)}}{{x - y}}dy}
\]
on Lorentz spaces on
R
+
{\mathbb {R}_ + }
with respect to the measure
4
x
d
x
{4^x}dx
. This is related to the harmonic analysis of radial functions on hyperbolic spaces. We prove that this operator is bounded on the Lorentz spaces
L
2
,
9
(
R
+
,
4
x
d
x
)
,
1
>
q
>
+
∞
{L^{2,9}}({\mathbb {R}_ + },{4^x}dx),1 > q > + \infty
, and it maps the Lorentz space
L
2
,
1
(
R
+
,
4
x
d
x
)
{L^{2,1}}({\mathbb {R}_ + },{4^x}dx)
into a space that we call WEAK-
L
2
,
1
(
R
+
,
4
x
d
x
)
{L^{2,1}}({\mathbb {R}_ + },{4^x}dx)
. We also prove that
H
\mathcal {H}
maps
L
1
(
R
+
,
4
x
d
x
)
{L^1}({\mathbb {R}_ + },{4^x}dx)
into WEAK-
L
1
(
R
+
,
4
x
d
x
)
+
L
2
(
R
+
,
4
x
d
x
)
{L^1}({\mathbb {R}_ + },{4^x}dx) + {L^2}({\mathbb {R}_ + },{4^x}dx)
.