Conditions for boundedness and compactness of product-convolution operators
g
→
P
h
C
f
g
=
h
⋅
(
f
∗
g
)
g \to {P_h}{C_f}g = h \cdot (f\ast g)
on spaces
L
p
(
G
)
{L_p}(G)
are studied. It is necessary for boundedness to define a class of "mixed-norm" spaces
L
(
p
,
q
)
(
G
)
{L_{(p,q)}}(G)
interpolating the
L
p
(
G
)
{L_p}(G)
spaces in a natural way
(
L
(
p
,
p
)
=
L
p
)
({L_{(p,p)}} = {L_p})
. It is then natural to study the operators acting between
L
(
p
,
q
)
(
G
)
{L_{(p,q)}}(G)
spaces, where
G
G
has a compact invariant neighborhood. The theory of
L
(
p
,
q
)
(
G
)
{L_{(p,q)}}(G)
is developed and boundedness and compactness conditions of a nonclassical type are obtained. It is demonstrated that the results extend easily to a somewhat broader class of integral operators. Several known results are strengthened or extended as incidental consequences of the investigation.