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

Abstract

For a given measurable space (Ω, Σ), and a vector measure m : Σ −→ X with values in a Banach space X we consider the spaces of p-power integrable and weakly integrable, respectively, functions with respect to the measure m, L p (m) and L p w (m), for 1 p < ∞.In this note we describe the real interpolated spaces that we obtain when the K -method is applied to any couple of these spaces.

Interpolation between weighted Lorentz spaces with respect to a vector measure

“…For the proof of above proposition you can see [14]. In the sequel, we prove the generalization of Steffensen's inequality for L p (∥m∥) spaces.…”

“…In [14] Fernandez et al introduced Lorentz spaces with respect to a vector measure and given some of their fundamental properties.…”

“…Integration of scalar functions with respect to a countably additive vector measure m : Σ → X with values in a Banach space X was introduced by Bartle-Dunford-schwartz [1] and studied by Klvanek-Knowles [18], and Lewis [19,20]. Recently, several papers have analysed the properties of the spaces of (weakly) p-integrable functions (L p w (m)) L P (m), these may be found in, for example, [8,[14][15][16][17]26].…”

“…The Calderón complex interpolation [X 0 , X 1 ] θ and [X 0 , X 1 ] θ , with 0 < θ < 1 of the couples (X 0 , X 1 ) where X 0 and X 1 are spaces L p (m) or L p w (m), with 1 ≤ p < ∞, were obtained in [16] and in [12] the Complex interpolation of Orlicz spaces with respect to a vector measure was identified. Moreover, 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 ≤ ∞, for vector measures on σ-algebras were studied in [14]. More precisely, Let 0 < θ < 1 ≤ q ≤ ∞, 1 ≤ p 0 < p 1 ≤ ∞, and…”

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

“…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.…”