Characterizations are obtained for those pairs of weight functions
w
,
υ
w,\upsilon
for which the Hardy operator
T
f
(
x
)
=
∫
0
x
f
(
s
)
d
s
Tf(x) = \int _0^x {f(s)\;ds}
is bounded from the Lorentz space
L
r
,
s
(
(
0
,
∞
)
,
υ
d
x
)
{L^{r,s}}((0,\infty ),\upsilon \,dx)
to
L
p
,
q
(
(
0
,
∞
)
,
w
d
x
)
,
0
>
p
,
q
,
r
,
s
⩽
∞
{L^{p,q}}((0,\infty ),w\,dx),0 > p,q,r,s \leqslant \infty
. The modified Hardy operators
T
η
f
(
x
)
=
x
−
η
T
f
(
x
)
{T_\eta }f(x) = {x^{ - \eta }}Tf(x)
for
η
\eta
real are also treated.