Interpolation of variable Hardy–Lorentz–Karamata spaces associated with rearrangement functions

In this article, we introduce variable Lorentz–Karamata spaces \({\mathcal {L}}_{p(\cdot ),q,b}(R)\) defined by rearrangement functions and develop the martingale theory in this framework. The real interpolation theory for variable Lorentz–Karamata spaces is presented. Based on this and the new atomic decomposition, we study the real interpolation theory for variable martingale Hardy–Lorentz–Karamata spaces. We also characterize the real interpolation spaces between variable martingale Hardy spaces and \(BMO_2\) spaces. The results obtained here generalize the previous results for variable Lorentz spaces as well as for variable martingale Hardy–Lorentz spaces. Moreover, we remove the condition \(\theta +p_->1\) in [Banach J. Math. Anal. 2023, 17(3): 47].