A sharp blow-up estimate for the Lebesgue norm

We prove that if p> 1 and ψ:] 0 , p- 1 [→] 0 , ∞[is nondecreasing, then sup0<1ψ(p-11-logt)‖f∗‖Lp(t,1)⇕ψ∈Δ2∩L∞.Here f is a Lebesgue measurable function on (0, 1) and f∗ denotes the decreasing rearrangement of f. The proof generalizes and makes sharp an equivalence previously known only in the particular case when ψ is a power; such case had a relevant role for the study of grand Lebesgue spaces. A number of consequences are discussed, among which: the behavior of the fundamental function of generalized grand Lebesgue spaces, an analogous equivalence in the case the assumption on the monotonicity of ψ is dropped, and an optimal estimate of the blow-up of the Lebesgue norms for functions in Orlicz–Zygmund spaces