Martingale inequalities in variable Lorentz–Karamata spaces

In this paper, we introduce a new class of function spaces, which unify and generalize Lorentz–Karamata spaces, variable Lorentz spaces, and other several classical function spaces. Based on the new spaces, we define five variable martingale Hardy–Lorentz–Karamata spaces and discuss the relationships among them. Our method is the atomic decomposition by some new techniques, which rigorously improves the known results in previous literature.