Some examples of equivalent rearrangement‐invariant quasi‐norms defined via f∗$f^*$ or f∗∗$f^{**}$

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