The John-Nirenberg space: Equality of the vanishing subspaces $VJN_p$ and $CJN_p$

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