POSITIVE MULTILINEAR OPERATORS ACTING ON WEIGHTED LP SPACES

For j = 1, 2,..., n + 1, let (Xj, ∑j, μj) be σ-finite measure spaces and let P(Xj) denote the class of nonnegative measurable functions on Xj. Given a positive multilinear operator M:P(X1) × P(X2) × ... ×P(Xn) → P(Xn+1), and fixed indices p1, p2,..., pn, q ε{lunate} [1, ∞], we consider the problem of determining those nonnegative (weight) functions w1, w2,..., wn and v on X1, X2,..., Xn and Xn + 1, respectively, for which ∫X [M(f1, f2 ..., fn) v]q dμn+1 1 q ≤C ∏ j=1 n ∫X (fjwj)p dμj 1 p, with C > 0 independent of fjε{lunate}P(Xj), j=1,2, ..., n. © 1992.