It is a subtle fact of the theory of regular operators on Banach lattices that every linear operator T : L1 (μ) → L1 (ν) is norm-bounded if and only if it is regular. We generalize this result to the setting of operators taking values in a Lebesgue–Bochner space. Our main result asserts that every linear operator T : L1(μ) → L1 (ν, X) is norm-bounded if and only if it is dominated. We show that this result is no longer true for Lebesgue–Bochner domain spaces. As a consequence of the main theorem, we obtain a generalized version of the Grothendieck inequality for linear norm-bounded operators from L1 (μ) to a Lebesgue–Bochner space L1 (ν, X).

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