Lions–Schechter's methods of complex interpolation and some quantitative estimates

Abstract

Some quantitative estimates concerning multi-dimensional rotundity, weak noncompactness, and certain spectral inequalities are formulated for Lions-Schechter's complex methods of interpolation with derivatives.The theory of the complex interpolation methods owes its origin to the famous interpolation theorem of Riesz-Thorin [3, Theorem 1.1.1]. Some fundamental inequalities due to Calderón [3, Lemma 4.3.2] imply that the boundedness of linear operators can be interpolated between Banach spaces with a logarithmically convex estimate for the norms of the interpolated operators. This motivated several authors to investigate the behavior of other properties of Banach spaces and linear operators under interpolation in both qualitative and quantitative way. For instance, Salvatori and Vignati obtained the estimate for multi-dimensional rotundity [11], Kryczka and Prus studied the measure of weak noncompactness [9], and Albrecht and Müller formulated some spectral inequalities under the complex interpolation methods [2].In the present paper, we treat the similar problems for the more general Lions-Schechter's methods of complex interpolation with derivatives. In the first section, we review the different variants of these methods and formulate some basic inequalities. Section 2 includes the estimate

Commutator Estimates for Interpolation Scales with Holomorphic Structure

“…Sp T, X (±1) ,s ⊆ Sp T, X ,s follows by this reiteration result and Proposition 3.2 in a similar way for the proof of [10,Prop. 3.2].…”

On interpolation of weakly compact bilinear operators

“…A quantitative approach based on the measure of weak noncompactness γ was undertaken by Kryczka, Prus and Szczepanik [29] and further developed by Kryczka and Prus [28], covering the case of complex interpolation of weakly compact operators. Other results in this direction can be seen in the papers by Fan [19] and Szwedek [40].…”