  We consider Lorentz-Karamata spaces, small and grand Lorentz-Karamata spaces,
and the so-called $\mathcal{L}$, $\mathcal{R}$, $\mathcal{LL}$, $\mathcal{RL}$,
$\mathcal{RL}$, and $\mathcal{RR}$ spaces. The quasi-norms for a function $f$
in each of these spaces can be defined via the non-increasing rearrangement
$f^*$ or via the maximal function $f^{**}$. 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.
