Real interpolation for variable martingale Hardy–Lorentz–Karamata spaces

In this paper, we study the real interpolation theory for variable Lorentz–Karamata spaces as well as for the corresponding martingale Hardy spaces. As a consequence, the generalization of Doob’s maximal inequality will be proved. Moreover, the atomic decompositions of the variable martingale Hardy–Lorentz–Karamata spaces are also presented. The results obtained here are new even for martingale Hardy–Lorentz–Karamata spaces.