## Abstract

We prove that if p>1 and ψ:]0,p−1]→]0,∞[ is just nondecreasing and differentiable (hence not necessarily Δ_{2}), then for every f Lebesgue measurable function on (0,1) sup0<ε<p−1ψ(ε)‖f‖_{Lp−ε(0,1)}≲sup0<t<1S_{ψ}(t)‖f^{⁎}‖_{Lp(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 Δ_{2} 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∈Δ_{2}.

Original language | English |
---|---|

Article number | 124550 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 493 |

Issue number | 2 |

DOIs | |

Publication status | Published - 15 Jan 2021 |

### Bibliographical note

Funding Information:The work of the third author was supported by the EPSRC grant EP/K02390X/1 and the Leverhulme grant RPG-2018-438 .

Funding Information:

The first author acknowledges support from the Austrian Science Fund (FWF) through the special research program Taming complexity in partial differential systems (Grant SFB F65 ).

Funding Information:

The first author acknowledges support from the Austrian Science Fund (FWF) through the special research program Taming complexity in partial differential systems (Grant SFB F65). All the authors acknowledge support from ESI, the Erwin Schr?dinger International Institute for Mathematics and Physics in Wien, given in occasion of the Workshop on New Trends in the Variational Modeling and Simulation of Liquid Crystals held at ESI, in Wien, on December 2-6, 2019. The work of the third author was supported by the EPSRC grant EP/K02390X/1 and the Leverhulme grant RPG-2018-438.

Publisher Copyright:

© 2020 Elsevier Inc.

## Keywords

- Banach function spaces
- Grand Lebesgue spaces
- Lebesgue spaces
- Norm blow-up
- Orlicz-Zygmund spaces
- Δ condition