Limiting interpolation spaces via extrapolation

TL;DR

This paper characterizes limiting interpolation spaces for the real method using extrapolation theory, introducing new operator boundedness criteria and extending classical reiteration theorems to unify and generalize previous results.

Contribution

It provides a complete characterization of limiting interpolation spaces via simple operator boundedness, extending classical theorems and unifying various special cases in the literature.

Findings

Characterization of limiting interpolation spaces using operator boundedness.

Extension of Holmstedt's reiteration theorem to limiting spaces.

Applications to Matsaev ideals, Grand Lebesgue spaces, and new vector-valued extrapolation theorems.

Abstract

We give a complete characterization of limiting interpolation spa\-ces for the real method of interpolation using extrapolation theory. For this purpose the usual tools (e.g., Boyd indices or the boundedness of Hardy type operators) are not appropriate. Instead, our characterization hinges upon the boundedness of some simple operators (e.g. $f\mapsto f(t^{2})/t$, or $f\mapsto f(t^{1/2}% )$) acting on the underlying lattices that are used to control the $K$- and $J$-functionals. Reiteration formulae, extending Holmstedt's classical reiteration theorem to limiting spaces, are also proved and characterized in this fashion. The resulting theory gives a unified roof to a large body of literature that, using ad-hoc methods, had covered only special cases of the results obtained here. Applications to Matsaev ideals, Grand Lebesgue spaces, Bourgain-Brezis-Mironescu-Maz'ya-Shaposhnikova limits, as well as a new vector valued extrapolation theorems, are provided.