On tensor products of operators from L ^p to L ^q

Abstract

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 Σ_i,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 Σ_i,j f _i ⊗ f _i fails to converge unconditionally.

This research was done at the Institute Des Hautes Etudes Scientifiques during the Fall of 1988. The authors wish to express their appreciation to the IHES for their warm hospitality and support. The first-named author also gratefully acknowledges the University of Texas at Austin for the Faculty Research Assignment which made this project possible. His work is partially supported by NSF DMS-8601752; the work of the second-named author is partially supported by NSF DMS-8702058.