The following question concerning the computation of the norms of the tensor products of operators in the Lebesgue spaces is studied: Is it true that the norm of the tensor product A ⊗ B : L^p (μ ⊗ μ) → L^q (ν ⊗ ν) of operators A : L^p(μ) → L^q(ν) and B : L^p(μ) → L ^q(ν) coincides with the product ∥A∥∥B∥ of their norms? An answer is positive if and only if l ≤ p ≤ q ≤ + ∞.