Real interpolation for adapted sequence spaces with variable exponents

Abstract

We study real interpolation for adapted sequence spaces with variable exponents. Let $(\hat{I}\text{c◯},\mathcal{F},\mathbb{P};({\mathcal{F}}_n{)}_{n\ge 1})$ be a filtered complete probability space, let $p(\cdot )\in \mathcal{P}(\hat{I}\text{c◯})$, and let $0<q\le \infty$ and $0<\hat{I}¸<1$. We prove that \[ \left(L^{\mathrm{ad}}_{p(\cdot)},L^{\mathrm{ad}}_{\infty}\right)_{θ,q} = L^{\mathrm{ad}}_{\widetilde{p}(\cdot),q}, \qquad \frac{1}{\widetilde{p}(\cdot)} = \frac{1-θ}{p(\cdot)}, \] with equivalent quasi-norms. Here $L_{\tilde{p}(\cdot ),q}^{\mathrm{ad}}$ consists of adapted sequences $f=(f_n{)}_{n\ge 1}$ whose square function $\ddot{I}ƒ(f)={\left({\sum }_{n=1}^{\infty }|f_n{|}^2\right)}^{1/2}$ 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.

15 pages, no figures