The Hardy-Littlewood theorem on fractional integration for Fourier series says that if
I
σ
g
∼
∑
n
≠
0
|
n
|
−
σ
g
^
(
n
)
e
int
{I_\sigma }g \sim \sum \nolimits _{n \ne 0} {|n{|^{ - \sigma }}\hat g} (n){e^{\operatorname {int} }}
, then
I
σ
{I_\sigma }
is bounded from
L
p
{L^p}
to
L
q
{L^q}
, where
1
>
p
>
q
>
∞
,
1
q
=
1
p
−
σ
1 > p > q > \infty ,\frac {1}{q} = \frac {1}{p} - \sigma
. We shall establish an analogue of this theorem for the Laguerre function system
{
(
n
!
Γ
(
n
+
α
+
1
)
)
1
2
L
n
α
(
x
)
e
−
x
2
x
α
2
}
n
=
0
∞
\left \{ {{{\left ( {\frac {{n!}}{{\Gamma (n + \alpha + 1)}}} \right )}^{\frac {1}{2}}}L_n^\alpha (x){e^{ - \frac {x}{2}}}{x^{\frac {\alpha }{2}}}} \right \}_{n = 0}^\infty
.