Oscillation functionals in rearrangement-invariant spaces yield subcritical, supercritical, and critical embeddings, with critical regimes producing logarithmic Hansson-type refinements via a deviation function of ψ/ϕ_X that recover and extend classical endpoint results including Trudinger-type ones
Masty lo, The modulus of smoothness in metric spaces and related problems, Potential Anal.35(2011), 301–328
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Oscillation Functionals and Embeddings in Rearrangement-Invariant Spaces
Oscillation functionals in rearrangement-invariant spaces yield subcritical, supercritical, and critical embeddings, with critical regimes producing logarithmic Hansson-type refinements via a deviation function of ψ/ϕ_X that recover and extend classical endpoint results including Trudinger-type ones