The John–Nirenberg Space: Equality of the Vanishing Subspaces \(VJN_p\) and \(CJN_p\)

The John–Nirenberg spaces \(JN_p\) $J{N}_p$ are generalizations of the space of bounded mean oscillation BMO with \(JN_\infty =BMO\) $J{N}_{\infty }=BMO$ . Their vanishing subspaces \(VJN_p\) $VJ{N}_p$ and \(CJN_p\) $CJ{N}_p$ are defined in similar ways as VMO and CMO, which are subspaces of BMO. As our main result, we prove that \(VJN_p\) $VJ{N}_p$ and \(CJN_p\) $CJ{N}_p$ coincide by showing that certain Morrey type integrals of \(JN_p\) $J{N}_p$ functions tend to zero for small and large cubes. We also show that \(JN_{p,q}(\mathbb {R}^n) =L^p(\mathbb {R}^n) / \mathbb {R}\) $J{N}_{p,q}({\mathbb{R}}^n)=L^p({\mathbb{R}}^n)/\mathbb{R}$ , if \(p = q\) $p=q$ .