It is known that for
0
>
p
≤
1
0>p\leq 1
, every linear operator
T
:
L
p
(
μ
)
→
L
p
(
λ
)
T: L_p( \mu ) \to L_p(\lambda )
is norm-bounded if and only if it is regular. Recently, this has been generalized to operators from
L
1
(
μ
)
L_1(\mu )
into the Lebesgue-Bochner space
L
1
(
λ
;
X
)
L_1(\lambda ;X)
in the form of dominated operators: every linear operator
T
:
L
1
(
μ
)
→
L
1
(
λ
;
X
)
T: L_1( \mu ) \to L_1(\lambda ;X)
is norm-bounded if and only if it is dominated. Using another method of proof, we generalize this result to all indices
0
>
p
≤
1
0>p\leq 1
. Our result asserts that if
X
X
is a
p
p
-Banach space, then every linear operator
T
:
L
p
(
μ
)
→
L
p
(
λ
;
X
)
T: L_p( \mu ) \to L_p(\lambda ;X)
is norm-bounded if and only if there is a positive operator
S
:
L
1
(
μ
)
→
L
1
(
λ
)
S:L_1(\mu ) \to L_1(\lambda )
satisfying for every
f
∈
L
p
(
μ
)
f\in L_p(\mu )
,
\[
‖
T
f
(
ω
)
‖
X
p
≤
S
(
|
f
|
p
)
(
ω
)
,
λ
-a.e
.
\|Tf(\omega )\|_X^p \leq S(|f|^p)(\omega ), \quad \text {$\lambda $-a.e}.
\]
We also obtain as a consequence, a version of Grothendieck inequality for bounded linear operators from
L
p
(
μ
)
L_p(\mu )
into
L
p
(
λ
;
X
)
L_p(\lambda ;X)
for
0
>
p
≤
1
0>p\leq 1
.