SCHUR MULTIPLIERS ON B(L-p , L-q)

Let (Omega(1), F-1, mu(1)) and Omega(1), F-2, mu(2)) be two measure spaces and 1 <= p, q <= +infinity. We give a definition of Schur multipliers on B(L-p(Omega(1)), L-q(Omega(2))) which extends the definition of classical Schur multipliers on B(l(p), l(q)). Our main result is a characterization of Schur multipliers in the case 1 <= q <= p +infinity. When 1 < q <= p < +infinity, phi is an element of L-infinity(Omega(1 )x Omega(2)) is a Schur multiplier on B(L-p(Omega(1)), L-q(Omega(2))) if and only if there are a measure space (a probability space when p not equal q) (Omega, mu), a is an element of L-infinity(mu(1), L-p(mu)) and b is an element of L-infinity(mu(2), L-q'(mu) such that, for almost every (s, t) is an element of Omega(1) x Omega(2,) phi(s,t) = < a(s),b(t)>. This result is new, even in the classical case. As a consequence, we give new inclusion relationships among the spaces of Schur multipliers on B (l(p), l(q)).