Real interpolation of variable martingale Hardy spaces and BMO spaces

Abstract In this paper, we mainly consider the real interpolation spaces for variable Lebesgue defined by decreasing rearrangement function and corresponding martingale Hardy spaces. Let $$0<q\le \infty $$ 0 < q ≤ ∞ $$0<\theta <1$$ θ 1 . Our three main results are following: $$\begin{aligned}{} & {} ({\mathcal {L}}_{p(\cdot )}({\mathbb {R}}^n),L_{\infty }({\mathbb {R}}^n))_{\theta ,q}={\mathcal {L}}_{{p(\cdot )}/(1-\theta ),q}({\mathbb {R}}^n),\\{} {H}}_{p(\cdot )}^s(\Omega ),H_{\infty }^s(\Omega ))_{\theta {H}}_{{p(\cdot ),q}^s(\Omega ) \end{aligned}$$ ( L p · ) R n , = / - H s Ω $$\begin{aligned} ),BMO_2(\Omega )}/(1- \theta ), B M O 2 where exponent $$p(\cdot )$$ is a measurable function.