Oscillation Functionals and Embeddings in Rearrangement-Invariant Spaces

TL;DR

This paper investigates how oscillation functionals influence embeddings in rearrangement-invariant spaces, revealing a classification into regimes and deriving explicit endpoint results with logarithmic and Trudinger-type refinements.

Contribution

It introduces a classification of embeddings based on oscillation functionals, analyzing the impact of space geometry and growth functions, and extends classical endpoint embeddings.

Findings

Identification of subcritical, supercritical, and critical regimes.

Derivation of logarithmic refinements in the critical regime.

Extension of classical endpoint embedding results.

Abstract

We study embeddings associated with oscillation functionals in rearrangement-invariant spaces. More precisely, given a positive function \(\psi\), we analyze how the interaction between the geometry of the underlying space and the growth of \(\psi\) determines the behaviour of these embeddings, leading to a natural classification into subcritical, supercritical and critical regimes. We prove that in the critical regime logarithmic refinements of Hansson type appear, governed by a deviation function associated with the quotient \(\psi/\varphi_X\), where \(\varphi_X\) is the fundamental function of the underlying space. This leads to explicit Hansson-type targets and, in the bounded case of the deviation function, to Trudinger-type consequences. The results recover and extend several classical endpoint embeddings.