New limiting real interpolation methods and their connection with the methods associated to the unit square

Abstract

over the unit square MSC (2010) Primary: 46B70; Secondary: 46E30Dedicated to Professor Hans Triebel on the occasion of his 75th birthday.We study limit K-spaces for general Banach couples, not necessarily ordered. They correspond to the extreme choice θ = 0, 1 in the realization of the real method as a K-space. We also show the connection of these limit spaces with interpolation methods defined by the unit square.

New limiting variants of the classical reiteration theorem for the K-interpolation method

“…It is well known that is bounded as a map from to and also as a map from to . Therefore, by the interpolation property of the K -interpolation method (see [ 13 , Proposition 3.2]), is bounded as a map from to . Thus, the proof is complete if we show that and .…”

“…Recently, Cobos et al [ 13 ] have defined two new scales of limiting K -interpolation spaces and , corresponding to the limiting values , without using the extra function b . Namely, and consist of elements with the following finite quasi-norms: and respectively.…”

“…Namely, and consist of elements with the following finite quasi-norms: and respectively. The main purpose in the paper [ 13 ] was to investigate the connection between these limiting K -interpolation methods and the interpolation over the unit square. Let us mention that these limiting K -interpolation methods were earlier considered in [ 14 ] and [ 15 ] in the case when the underlying couple is ordered, meaning that (or ) is continuously embedded in (or ).…”

General Holmstedt’s Formulae for theK-Functional

“…In recent papers [1][2][3][4][5][6][7][8][9][10][11] the classical Holmstedt formula for the -functional [12] was extended to more general cases. See also [13] for more results about generalized Holmstedt's formula and reiteration theorems not only for the -method, but for the -method as well.…”

Reiteration Theorems for Two-Parameter Limiting Real Interpolation Methods

“…The motivation for introducing the two-parameter limiting spaces ( 0 , 1 ) { , } mainly stems from the fact the sum of the limiting spaces ( 0 , 1 ) 0, ; and ( 0 , 1 ) 1, ; , introduced by Cobos et al [11] in connection with the interpolation over the unit square, is precisely ( 0 , 1 ) , . This fact is established in [9,Proposition 3.4] for = , and the same argument also works for arbitrary values of and .…”