Some remarks on interpolation of families of quasi-Banach spaces

Summary

Abstract. We study some questions raised in theory of complex interpolation of quasi-Banach spaces. In particular we give a criterion for the interpolated space to be locally convex.

1. Introduction. In [1] and [2], Coifman, Cwikel, Rochberg, Saghar and Weiss introduced and studied complex interpolation of families of Banach spaces. Recently, Tabacco [11],[12] and Rochberg [10] have studied the extension of these ideas to the non-locally convex quasi-Banach case.

We let T denote the unit circle in the complex plane and λ denote normalizecf Haar measure on T, i.e. dλ = (2π)⁻¹dθ. Δ denotes the unit disk, {z : ∣z∣ < 1}. We then suppose that we are given a family of quasinormed spaces X_w for w ∊ T and define interpolation spaces X_z for z ∊ Δ. The precise details of the construction are given in Section 2.

In this paper we prove two main results on interpolation of analytic families of quasi-Banach spaces. In Theorem 4, we answer a question of Rochberg [10] by giving a condition for the interpolated space to be locally convex. We use here the notion of (Rademacher) type. A quasi-Banach space X is of type p where 0 < p ≤ 2 if there is a constant C so that if x₁,…, x_n ∊ X then

where the signs ∊_k = ±1 are chosen at random. In fact if p < 1 then type p is equivalent to p-normability [5], but there are type one spaces which are not locally convex (e.g. the Lorentz spaces where 1 < p < ∞, or the Ribe space [5]).