Let
G
G
be a locally compact unimodular group equipped with Haar measure
m
m
,
G
^
\hat G
its unitary dual and
μ
\mu
the Plancherel measure (or something closely akin to it) on
G
^
\hat G
. When
G
G
is a euclidean motion group, a non-compact semisimple Lie group or one of the Heisenberg groups we prove local uncertainty inequalities of the following type: given
θ
∈
[
0
,
1
2
)
\theta \in \left [ {0,\tfrac {1} {2}} \right .)
there exists a constant
K
θ
{K_\theta }
such that for all
f
f
in a certain class of functions on
G
G
and all measurable
E
⊆
G
^
E \subseteq \hat G
,
\[
(
∫
E
Tr
(
π
(
f
)
∗
π
(
f
)
)
d
μ
(
π
)
)
1
/
2
⩽
K
θ
μ
(
E
)
θ
|
|
ϕ
θ
f
|
|
2
{\left ( {\int _E {\operatorname {Tr} (\pi {{(f)}^{\ast }}\pi (f))\,d\mu (\pi )} } \right )^{1/2}} \leqslant {K_\theta }\mu {(E)^\theta }||{\phi _\theta }f|{|_2}
\]
where
ϕ
θ
{\phi _\theta }
is a certain weight function on
G
G
(for which an explicit formula is given). When
G
=
R
k
G = {{\mathbf {R}}^k}
the inequality has been established with
ϕ
θ
(
x
)
=
|
x
|
k
θ
{\phi _\theta }(x) = |x{|^{k\theta }}
.