We give sufficient conditions for inequalities of the form
\[
(
∫
(
∫
G
(
x
−
y
)
f
(
y
)
d
μ
(
y
)
)
q
d
ω
(
x
)
)
1
/
q
⩽
C
(
∫
|
f
(
y
)
|
p
d
ν
(
y
)
)
1
/
p
{\left ( {\int {{{\left ( {\int {G(x - y)f(y)\,d\mu (y)} } \right )}^q}\,d\omega (x)} } \right )^{1/q}}\, \leqslant C{\left ( {\int {|f(y){|^p}d\nu (y)} } \right )^{1/p}}
\]
to hold for measurable functions
f
f
. We determine the dependence of the constant
C
C
on the measures
μ
\mu
,
ν
\nu
,
ω
\omega
and give some applications.