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