Real interpolation for adapted sequence spaces with variable exponents

We study real interpolation for adapted sequence spaces with variable exponents. Let $(\mathrm{\Omega },\mathcal{F},\mathbb{P};({\mathcal{F}}_n{)}_{n\ge 1})$ be a filtered complete probability space, let $p(\cdot )\in \mathcal{P}(\mathrm{\Omega })$, and let $0<q\le \mathrm{\infty }$ and $0<\theta <1$. We prove that $${(L_{p(\cdot )}^{\mathrm{a}\mathrm{d}},L_{\mathrm{\infty }}^{\mathrm{a}\mathrm{d}})}_{\theta ,q}=L_{\tilde{p}(\cdot ),q}^{\mathrm{a}\mathrm{d}},\frac{1}{\tilde{p}(\cdot )}=\frac{1-\theta }{p(\cdot )},$$ with equivalent quasi-norms. Here $L_{\tilde{p}(\cdot ),q}^{\mathrm{a}\mathrm{d}}$ consists of adapted sequences $f=(f_n{)}_{n\ge 1}$ whose square function $\sigma (f)={\left({\sum }_{n=1}^{\mathrm{\infty }}\mathrm{\mid }{f}_n{\mathrm{\mid }}^2\right)}^{1\mathrm{/}2}$ belongs to the variable Lorentz space $L_{\tilde{p}(\cdot ),q}$. The proof uses a decomposition that preserves adaptedness and provides an upper estimate for the corresponding $K$-functional. No continuity condition on the variable exponent and no measurability relation between $p(\cdot )$ and the filtration are required.