Let
G
G
be a semi-simple Lie group in the Harish-Chandra class with maximal compact subgroup
K
K
. Let
Ω
K
\Omega _K
be minus the radial Casimir operator. Let
1
4
dim
(
G
/
K
)
>
S
G
>
1
2
dim
(
G
/
K
)
,
s
∈
(
0
,
S
G
]
\frac {1}{4} \dim (G/K) > S_G > \frac {1}{2} \dim (G/K) , s \in (0, S_G]
and
p
∈
(
1
,
∞
)
p \in (1,\infty )
be such that
\[
|
1
p
−
1
2
|
>
s
2
S
G
.
\left | \frac {1}{p} - \frac {1}{2} \right | > \frac {s}{2 S_G}.
\]
Then, there exists a constant
C
G
,
s
,
p
>
0
C_{G,s,p} >0
such that for every
m
∈
L
∞
(
G
)
∩
L
2
(
G
)
m \in L^\infty (G) \cap L^2(G)
bi-
K
K
-invariant with
m
∈
D
o
m
(
Ω
K
s
)
m \in Dom(\Omega _K^s)
and
Ω
K
s
(
m
)
∈
L
2
S
G
/
s
(
G
)
\Omega _K^s(m) \in L^{2S_G/s}(G)
we have,
‖
T
m
:
L
p
(
G
^
)
→
L
p
(
G
^
)
‖
≤
C
G
,
s
,
p
‖
Ω
K
s
(
m
)
‖
L
2
S
G
/
s
(
G
)
,
\begin{equation} \Vert T_m: L^p(\widehat {G}) \rightarrow L^p(\widehat {G}) \Vert \leq C_{G, s,p} \Vert \Omega _K^s(m) \Vert _{L^{2S_G/s}(G)}, \end{equation}
where
T
m
T_m
is the Fourier multiplier with symbol
m
m
acting on the non- commutative
L
p
L^p
-space of the group von Neumann algebra of
G
G
. This gives new examples of
L
p
L^p
-Fourier multipliers with decay rates becoming slower when
p
p
approximates
2
2
.