K-ENVELOPES FOR REAL INTERPOLATION METHODS

Abstract

Abstract. In this paper, we study the K-envelopes of the real interpolation methods with function space parameters in the sense of Brudnyi and Kruglyak [Y. A. Brudnyi and N. Ja. Kruglyak, Interpolation functors and interpolation spaces (NorthHolland, Amsterdam, Netherlands, 1991)]. We estimate the upper bounds of the Kenvelopes and the interpolation norms of bounded operators for the K -methods in terms of the fundamental function of the rearrangement invariant space related to the function space parameter . The results concerning the quasi-power parameters and the growth/continuity envelopes in function spaces are obtained. studied this connection and found some estimates of the growth/continuity envelopes in the framework of the classical real interpolation with the numerical parameter θ ∈ (0, 1).Our goal in this paper is to investigate the K-envelopes of the real interpolation methods with function space parameters in the sense of Brudnyi and Kruglyak [3]. In the first section, we give some preliminary information concerning the K interpolation methods and related topics. In Section 2, we study the K-envelopes and interpolation operators for the K -methods, and estimate the upper bounds in terms of the fundamental function of the rearrangement invariant (r.i.) space K ( L 1 , L ∞ ). Section 3 concerns with the K-envelopes and some properties of the K -methods, especially Lions-Peetre's methods of constants and means, with quasi-power parameters. In the final section, we apply these estimates on the growth/continuity envelopes in function spaces, and carry over the results in [9] to the setting of K -methods. Preliminaries.Let X = (X 0 , X 1 ) be a Banach couple with X = X 0 ∩ X 1 and X = X 0 + X 1 , and let X be an intermediate space for X. We denote by X 0 the regularisation of X for X, by X the Banach space dual of X 0 , and we write X = X 0 , X 1 as the dual couple of X. The notation B (X, Y ) (resp., B X, Y ) stands for the available at https://www.cambridge.org/core/terms. https://doi