Orlicz-Lorentz-Karamata Hardy martingale spaces: inequalities and fractional integral operators

Let \(0<q\le \infty \), b be a slowly varying function and \( \Phi : [0,\infty ) \longrightarrow [0,\infty ) \) be an increasing function with \(\Phi (0)=0\) and \(\lim \limits _{r \rightarrow \infty }\Phi (r)=\infty \). In this paper, we introduce a new class of function spaces \(L_{\Phi ,q,b}\) which unify and generalize the Lorentz-Karamata spaces with \(\Phi (t)=t^p\) and the Orlicz-Lorentz spaces with \(b\equiv 1\). Based on the new spaces, we introduce five new Hardy spaces containing martingales, the so-called Orlicz-Lorentz-Karamata Hardy martingale spaces and then develop a theory of these martingale Hardy spaces. To be precise, we first investigate several properties of Orlicz-Lorentz-Karamata spaces and then present Doob’s maximal inequalities by using Hardy’s inequalities. The characterization of these Hardy martingale spaces are constructed via the atomic decompositions. As applications of the atomic decompositions, martingale inequalities and the relation of the different martingale Hardy spaces are presented. The dual theorems and a new John-Nirenberg type inequality for the new framework are also established. Moreover, we study the boundedness of fractional integral operators on Orlicz-Lorentz-Karamata Hardy martingale spaces. The results obtained here generalize the previous results for Lorentz-Karamata Hardy martingale spaces as well as for Orlicz-Lorentz Hardy martingales spaces. Especially, we remove the condition that b is non-decreasing as in [38, 39] and the condition \(q_{\Phi ^{-1}}<1/q\) in [24], respectively.