Let μ, μ′, v, v’ denote σ-finite measures on certain measurable spaces, let 1 ≤ q,p ≤ ∞, and let S : L p (μ) → L q (μ′), T: L p (v) → L q (v’) be given bounded linear operators. It is proved that if q < p, then (in general) S ⊗ T extends to a bounded linear operator U from L p (μ × v) to L q (μ′ × v’) if and only if 1 ≤ q ≤ 2 ≤ p ≤ ∞. In this case there are (best) constants C p,q depending only on p and q with ‖U‖ ≤ C p,q ‖S‖ ‖T‖. These constants are all larger than one. They are computed in certain cases. For example, C 2,1 = C ∞,2 = √σ/2. (This contrasts with a lemma due to W. Beckner: if p ≤ q, then S ⊗ T always extends to such a U with ‖U‖ = ‖S‖ ‖T‖.) It is deduced as acorollary that if Σ f i and Σ g j are unconditionally converging series in L p (μ) and L p (v), then Σi,j f i ⊗ g j unconditionally converges in L p (μ × v) provided 1 ≤ p ≤ 2. However for every 2 < p < ∞ there is an unconditionally converging series Σ f i in l p so that Σi,j f i ⊗ f i fails to converge unconditionally.