### 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 |

### Keywords

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

## Cite this

*Journal of Mathematical Analysis and Applications*,

*493*(2), [124550]. https://doi.org/10.1016/j.jmaa.2020.124550