Multilinear Interpolation between Adjoint Operators

Multilinear interpolation is a powerful tool used in obtaining strong type boundedness for a variety of operators assuming only a finite set of restricted weak-type estimates. A typical situation occurs when one knows that a multilinear operator satisfies a weak L estimate for a single index q (which may be less than one) and that all the adjoints of the multilinear operator are of similar nature, and thus they also satisfy the same weak L estimate. Under this assumption, in this expository note we give a general multilinear interpolation theorem which allows one to obtain strong type boundedness for the operator (and all of its adjoints) for a large set of exponents. The key point in the applications we discuss is that the interpolation theorem can handle the case q ≤ 1. When q > 1, weak L has a predual, and such strong type boundedness can be easily obtained by duality and multilinear interpolation (c.f. [1], [5], [7], [12], [14]). 1. Multilinear operators We begin by setting up some notation for multilinear operators. Let m ≥ 1 be an integer. We suppose that for 0 ≤ j ≤ m, (Xj, μj) are measure spaces endowed with positive measures μj. We assume that T is an m-linear operator of the form T (f1, . . . , fm)(x0) := ∫