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

Abstract

We consider Lorentz-Karamata spaces, small and grand Lorentz-Karamata spaces, and the so-called , , , , , and  spaces. The quasi-norms for a function 𝑓 in each of these spaces can be defined via the nonincreasing 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 an application, we demonstrate how our general results can be used to establish interpolation formulae for grand and small Lorentz-Karamata spaces.