A sharp blow-up estimate for the Lebesgue norm

Abstract

We prove that if $p>1$ and $\psi :]0,p-1[\rightarrow ]0,\infty [$ is nondecreasing, then

$$\begin{aligned} \sup _{0<\varepsilon<p-1} \psi (\varepsilon ) \Vert f\Vert _{L^{p-\varepsilon }(0,1)}\approx & {} \sup _{0<t<1} \psi \left( \frac{p-1}{1-\log t}\right) \Vert f^*\Vert _{L^{p}(t,1)} \\&\mathop {\Updownarrow }\limits _{{\begin{array}{c} \psi \,\in \,\Delta _2\,\cap \, L^\infty . \end{array}}} \end{aligned}$$

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 $\psi$ 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 $\psi$ is dropped, and an optimal estimate of the blow-up of the Lebesgue norms for functions in Orlicz–Zygmund spaces.