A sharp blow-up estimate for the Lebesgue norm

We prove that if p>1 is psi :] 0, p - 1[->] 0,8[ is nondecreasing,, then sup(0<epsilon p-1) psi(epsilon)parallel to f parallel to(L)p-epsilon (0,1) approximate to sup(0<t<1) psi(p - 1/1-logt) parallel to f*parallel to L-p(t,1). 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 ? 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 ? is dropped, and an optimal estimate of the blow-up of the Lebesgue norms for functions in Orlicz-Zygmund spaces