Garsia–Rodemich spaces: Local maximal functions and interpolation

We characterize the Garsia–Rodemich spaces $ \operatorname {GaRo}_{X}$ associated with a rearrangement invariant space via local maximal operators. Let $Q_{0}$ be a cube in $\mathbb {R}^{n}$. We show that there exists $s_{0}\in (0,1)$ such that for all $0