Interpolation of n Pairs and Counterexamples Employing Indices

The purpose of this paper is to make two modest contributions to the theory of interpolation of operators on function spaces. The first deals with the modification of interpolation theorems to handle interpolation of n pairs. This work was partially begun in [S, 91 where strong type interpolation theorems were obtained using techniques easily altered to fit the n pair situation. Section 2 presents a weak type theory of interpolation of tz pairs of spaces using the Calderdn operator S’, . In order to show that S, is a “maximal” weak type operator, the proof in [6] for n = 2 is simplified and extended. Section 3 deals with counterexamples to interpolation theorems which make use of indices derived from the fundamental functions of the spaces in the interpolation scheme. We show that interpolation theorems of Semenov [5] and Zippin [II] are incorrect by using a space furnished by Shimogaki [lo]. Under a suitable hypothesis a weak type theorem involving the fundamental indices does hold, but follows from a well-known result of Boyd [I] without employing the main ideas set forth in [5] and [3].