Oscillation Functionals and Embeddings in Rearrangement-Invariant Spaces

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 $\psi \mathrm{/}{\phi }_X$, where ${\phi }_X$ 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.