Real interpolation for adapted sequence spaces with variable exponents

We study real interpolation for adapted sequence spaces with variable exponents. Let $(\Omega,\mathcal{F},\mathbb{P};(\mathcal{F}_n)_{n\geq 1})$ be a filtered complete probability space, let $p(\cdot)\in\mathcal{P}(\Omega)$, and let $0<q\leq\infty$ and $0<\theta<1$. We prove that $$\left(L^{\mathrm{ad}}_{p(\cdot)},L^{\mathrm{ad}}_{\infty}\right)_{\theta,q} = L^{\mathrm{ad}}_{\widetilde{p}(\cdot),q}, \qquad \frac{1}{\widetilde{p}(\cdot)} = \frac{1-\theta}{p(\cdot)},$$ with equivalent quasi-norms. Here $L^{\mathrm{ad}}_{\widetilde{p}(\cdot),q}$ consists of adapted sequences $f=(f_n)_{n\geq 1}$ whose square function $\sigma(f)=\left(\sum_{n=1}^{\infty}|f_n|^2\right)^{1/2}$ belongs to the variable Lorentz space $L_{\widetilde{p}(\cdot),q}$. 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.