Conditions on nonnegative weight functions
u
(
x
)
u(x)
and
υ
(
x
)
\upsilon (x)
are given which ensure that an inequality of the form
(
∫
|
T
f
(
x
)
|
q
u
(
x
)
d
x
)
1
/
q
⩽
C
(
∫
|
f
(
x
)
|
p
υ
(
x
)
d
x
)
1
/
p
{(\smallint {\left | {Tf(x)} \right |^q}u(x)\;dx)^{1/q}} \leqslant C{(\smallint {\left | {f(x)} \right |^p}\upsilon (x)\;dx)^{1/p}}
holds for
1
⩽
q
>
p
>
∞
1 \leqslant q > p > \infty
, where
T
T
is an integral operator of the form
∫
−
∞
x
K
(
x
,
y
)
f
(
y
)
d
y
\int _{ - \infty }^x {K(x,y)f(y)dy}
or
∫
x
∞
K
(
y
,
x
)
f
(
y
)
d
y
\int _x^\infty {K(y,x)f(y)\;dy}
and
C
C
a constant independent of
f
f
. Specifically a number of inequalities for well-known classical operators are obtained. Inequalities of the above form for
1
⩽
p
⩽
q
>
∞
1 \leqslant p \leqslant q > \infty
were obtained in [1].