Let
G
G
be a stratified, nilpotent Lie group and let
L
L
be a homogeneous sublaplacian on
G
G
. Let
E
(
λ
)
E(\lambda )
denote the spectral resolution of
L
L
on
L
2
(
G
)
{L^2}(G)
. Given a function
K
K
on
R
+
\mathbf {R}^+
, define the operator
T
K
{T_K}
on
L
2
(
G
)
{L^2}(G)
by
T
k
f
=
∫
0
∞
K
(
λ
)
d
E
(
λ
)
f
{T_k}f = \int _0^\infty \, {K(\lambda )\;dE(\lambda )\,f}
. Sufficient conditions on
K
K
to imply that
T
K
{T_K}
is bounded on
L
1
(
G
)
{L^1}(G)
and the maximal operator
K
∗
φ
(
x
)
=
sup
t
>
0
|
T
K
t
φ
(
x
)
|
K^{\ast } \varphi (x) = \sup _{t > 0}|{T_{K_t}}\varphi (x)|
(where
K
t
(
λ
)
=
K
(
t
λ
)
{K_t}(\lambda ) = K(t\lambda )
) is of weak type
(
1
,
1
)
(1,1)
are given. Picking a basis
e
0
,
e
1
,
…
{e_0},{e_1},\ldots
of
L
2
(
G
/
Γ
)
{L^2}(G/\Gamma )
(
Γ
\Gamma
being a discrete cocompact subgroup of
G
G
) consisting of eigenfunctions of
L
L
, we obtain almost everywhere and norm convergence of various summability methods of
Σ
(
φ
,
e
j
)
e
j
,
φ
∈
L
p
(
G
/
Γ
)
,
1
⩽
p
>
∞
\Sigma (\varphi ,{e_j}){e_j},\varphi \in {L^p}(G/\Gamma ), 1 \leqslant p > \infty
.