Calder\'on-Mityagin couples of Banach spaces related to decreasing functions

TL;DR

This paper characterizes Calderón-Mityagin couples among Banach spaces related to decreasing functions, showing their interpolation structures mirror those of rearrangement invariant spaces and extending results to general measures.

Contribution

It establishes a correspondence between Calderón-Mityagin couples of decreasing function spaces and rearrangement invariant spaces, and extends interpolation results to general Borel measures.

Findings

Calderón-Mityagin couples are characterized among decreasing function spaces.

Interpolation structures of these spaces parallel those of rearrangement invariant spaces.

Results extend to spaces defined with general Borel measures on R.

Abstract

A number of Calder\'on-Mityagin couples and relative Calder\'on-Mityagin pairs are identified among Banach function spaces defined in terms of the least decreasing majorant construction on the half line. The interpolation structure of such spaces is shown to closely parallel that of the rearrangement invariant spaces, and it is proved that a couple of these spaces is a Calder\'on-Mityagin couple if and only if the corresponding couple of rearrangement invariant spaces is a Calder\'on-Mityagin couple. Consequently, the class of all interpolation spaces for any couple of spaces of this type admits a complete description by the K-method if and only if the class of all interpolation spaces for the corresponding couple of rearrangement invariant spaces does. Analogous results are proved for spaces defined in terms of the level function construction. In the main, the conclusions for both types of spaces remain valid when Lebesgue measure on the half line is replaced by a general Borel measure on R. However, for certain measures the class of interpolation spaces of these new spaces may be degenerate, reducing the "if and only if" of the main results to a single implication.