Real interpolation of variable martingale Hardy spaces and BMO spaces

In this paper, we mainly consider the real interpolation spaces for variable Lebesgue spaces defined by the decreasing rearrangement function and for the corresponding martingale Hardy spaces. Let 0<q≤∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<q\le \infty $$\end{document} and 0<θ<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\theta <1$$\end{document}. Our three main results are the following: (Lp(·)(Rn),L∞(Rn))θ,q=Lp(·)/(1-θ),q(Rn),(Hp(·)s(Ω),H∞s(Ω))θ,q=Hp(·)/(1-θ),qs(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}{} & {} ({\mathcal {L}}_{p(\cdot )}({\mathbb {R}}^n),L_{\infty }({\mathbb {R}}^n))_{\theta ,q}={\mathcal {L}}_{{p(\cdot )}/(1-\theta ),q}({\mathbb {R}}^n),\\{} & {} ({\mathcal {H}}_{p(\cdot )}^s(\Omega ),H_{\infty }^s(\Omega ))_{\theta ,q}={\mathcal {H}}_{{p(\cdot )}/(1-\theta ),q}^s(\Omega ) \end{aligned}$$\end{document}and (Hp(·)s(Ω),BMO2(Ω))θ,q=Hp(·)/(1-θ),qs(Ω),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} ({\mathcal {H}}_{p(\cdot )}^s(\Omega ),BMO_2(\Omega ))_{\theta ,q}={\mathcal {H}}_{{p(\cdot )}/(1- \theta ),q}^s(\Omega ), \end{aligned}$$\end{document}where the variable exponent p(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\cdot )$$\end{document} is a measurable function.