A multilinear Zafran theorem for the measure of noncompactness of operators

Abstract

We establish a multilinear analogue of Zafran's interpolation theorem for operators acting on products of quasi-Banach spaces generated by the general real methods. We consider multilinear operators between quasi-Banach couples and obtain interpolation estimates expressed in terms of the fundamental functions of the underlying sequence spaces. As an application, we study the measure of noncompactness of interpolated multilinear operators in this setting and derive a corresponding interpolation estimate for this quantity. In particular, we obtain a one-sided compactness result for interpolated multilinear operators acting on spaces associated with the abstract real methods. These results recover known bilinear theorems as special cases and extend earlier results on multilinear interpolation and on the interpolation of the measure of noncompactness.