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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 thesequasi-norms are equivalent. Most of the proofs are based on Hardy-typeinequalities. As application we demonstrate how our general results can be usedto establish interpolation formulae for the grand and small Lorentz-Karamataspaces.

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