The John-Nirenberg space: Equality of the vanishing subspaces $VJ{N}_p$ and $CJ{N}_p$

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