Some examples of equivalent rearrangement-invariant quasi-norms defined via f* or f**

Abstract

We consider Lorentz-Karamata spaces, small and grand Lorentz-Karamata spaces, and the so-called $\mathcal{L}$, $\mathcal{R}$, $\mathcal{LL}$, $\mathcal{R}\mathcal{L}$, $\mathcal{R}\mathcal{L}$, and $\mathcal{RR}$ spaces. The quasi-norms for a function $f$ in each of these spaces can be defined via the non-increasing rearrangement $f^{\ast }$ or via the maximal function $f^{\ast \ast }$. We investigate when these quasi-norms are equivalent. Most of the proofs are based on Hardy-type inequalities. As application we demonstrate how our general results can be used to establish interpolation formulae for the grand and small Lorentz-Karamata spaces.