Generalization of the Ehrling inequality and universal characterization of completely continuous operators

TL;DR

This paper extends Ehrling inequalities to more general operators, introducing Ehrling continuity and characterizing completely continuous operators through generalized inequalities, linking topology and operator theory.

Contribution

It introduces Ehrling continuity, provides a necessary and sufficient condition for it, and characterizes completely continuous operators via generalized Ehrling inequalities.

Findings

Ehrling continuity is characterized by a topological condition.

Complete continuity is linked to generalized Ehrling inequalities.

A universal relation among continuity, weak topology, and inequalities is established.

Abstract

The present work is devoted to an extension of the well-known Ehrling inequalities, which quantitatively characterize compact embeddings of function spaces, to more general operators. Firstly, a modified notion of continuity for linear operators, named \emph{Ehrling continuity} and inspired by the classical Ehrling inequality, is introduced, and then, a necessary and sufficient condition for Ehrling continuity is provided via arguments based on general topology. Secondly, general completely continuous operators between normed spaces are characterized in terms of (generalized) Ehrling type inequalities. To this end, the well-known local metrization of the weak topology (so to speak, a \emph{very weak norm}) plays a crucial role. Thanks to these results, a universal relation is observed among complete continuity, the very weak norm and generalized Ehrling type inequality.