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arXiv · 2604.05804

Oscillation Functionals and Embeddings in Rearrangement-Invariant Spaces

Joaquim Martin
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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.

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Joaquim Martin. 2026-04-07. Oscillation Functionals and Embeddings in Rearrangement-Invariant Spaces. https://arxiv.org/abs/2604.05804

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