An estimate of the blow-up of Lebesgue norms in the non-tempered case

We prove that if p>1 and ψ:]0,p−1]→]0,∞[ is just nondecreasing and differentiable (hence not necessarily Δ₂), then for every f Lebesgue measurable function on (0,1) sup0<ε<p−1⁡ψ(ε)‖f‖_{L^p−ε(0,1)}≲sup0<t<1⁡S_ψ(t)‖f^⁎‖_L^p(t,1), where f^⁎ denotes the decreasing rearrangement of f and S_ψ is defined, for ε∈]0,p−1[, through [Formula presented] where c_ψ is the normalizing constant chosen so that ν((p−1)−)=1. If ψ is in a class of functions satisfying the Δ₂ condition, essentially characterized by the so-called ∇^′ condition, then inequality (⁎) is sharp, in the sense that both sides are equivalent. Estimate (⁎) generalizes an inequality of the type obtained by the second author with Farroni and Giova in [6] under the growth condition ψ∈Δ₂.