Complex interpolation of spaces of integrable functions with respect to a vector measure

Abstract

Let (Ω, Σ) be a measurable space and m : Σ → X be a vector measure with values in the complex Banach space X. We apply the Calderón interpolation methods to the family of spaces of scalar p−integrable functions with respect to m with 1 ≤ p ≤ ∞. Moreover we obtain a result about the relation between the complex interpolation spaces[θ] for a Banach couple of interpolation (X 0 , X 1 ) such that X 1 ⊂ X 0 with continuous inclusion.

Complex interpolation of Orlicz spaces with respect to a vector measure

“…Corollary 4.2 and Theorem 4.5 extend the corresponding results given in [7] on spaces of p-integrable and weakly p-integrable functions to the setting of Orlicz spaces.…”

“…Associated to m , there are the Banach function spaces and , with , of (equivalence classes of ) scalar p ‐integrable and weakly p ‐integrable functions with respect to m . The Calderón interpolation spaces and (see and ) of the couples where X 0 and X 1 are spaces or were obtained in . In such a case, the first method always gives another ‐space and the second one yields an ‐space.…”

Real interpolation method on spaces of scalar integrable functions with respect to vector measures

“…These spaces has been studied in [6]. Moreover, the two Calderón's complex interpolation spaces, [X 0 , X 1 ] [θ] and [X 0 , X 1 ] [θ] , with 0 < θ < 1, of the complex Banach lattices couple (X 0 , X 1 ), where X 0 and X 1 are the any of the spaces L p (m) or L p w (m), with 1 p < ∞, were obtained in [7]. The purpose of this paper is to obtain the real interpolation spaces (X 0 , X 1 ) θ,q , where 0 < θ < 1 q ∞, and X 0 and X 1 are, as above, L p (m) or L p w (m), with 1 p ∞.…”

Duality theory for p-th power factorable operators and kernel operators.

“…[70]). The second complex method does not seem to give us more information since for the interpolation of p-integrable function spaces with respect to a vector measure, both methods coincide to each other (see [40]). …”

“…Chapter 5 contains two parts. The first one deals with some applications in the setting of the interpolation theory for Banach function spaces, that has been recently developed specifically for the case of the spaces of p-integrable functions with respect to a vector measure (see [26,36,37,40]). In particular, we show that the properties of p-th and (p, q)-th power factorability are inherited by Calderón complex interpolation.…”