Sobolev embedding into BMO and weak-L∞ for 1-dimensional probability measure

We characterize rearrangement invariant spaces X with respect to a suitable 1-dimensional probability μ (e.g. log-concave measure) such that the Sobolev embedding‖u‖BMO(R,μ)≤C(‖u′‖X+‖u‖L1(R,μ)) holds for any function u∈L1(R,μ), whose real-valued weakly derivative u′ belongs to X. Here BMO(R,μ) is the space of functions with bounded mean oscillation with respect to μ. We investigate the embedding in weak-L∞(R,μ), too.