Consider the space
X
=
(
0
,
∞
)
X=(0,\infty )
equipped with the Euclidean distance and the measure
d
μ
α
(
x
)
=
x
α
d
x
d\mu _\alpha (x)=x^{\alpha }dx
where
α
∈
(
−
1
,
∞
)
\alpha \in (-1,\infty )
is a fixed constant and
d
x
dx
is the Lebesgue measure. Consider the Laguerre operator
L
=
−
d
2
d
x
2
−
α
x
d
d
x
+
x
2
\displaystyle L=-\frac {d^2}{dx^2} -\frac {\alpha }{x}\frac {d}{dx}+x^2
on
X
X
. The aim of this article is threefold. Firstly, we establish a Calderón reproducing formula using a suitable distribution of the Laguerre operator. Secondly, we study certain properties of the Laguerre operator such as a Harnack type inequality on the solutions and subsolutions of Laplace equations associated to Laguerre operators. Thirdly, we establish the theory of the weighted homogeneous Besov and Triebel-Lizorkin spaces associated to the Laguerre operator. We define the weighted homogeneous Besov and Triebel-Lizorkin spaces by the square functions of the Laguerre operator, then show that these spaces have an atomic decomposition. We then study the fractional powers
L
−
γ
,
γ
>
0
L^{-\gamma }, \gamma >0
, and show that these operators map boundedly from one weighted Besov space (or one weighted Triebel-Lizorkin space) to another suitable weighted Besov space (or weighted Triebel-Lizorkin space). We also show that in particular cases of the indices, our new weighted Besov and Triebel-Lizorkin spaces coincide with the (expected) weighted Hardy spaces, the weighted
L
p
L^p
spaces or the weighted Sobolev spaces in Laguerre settings.