The John–Nirenberg Space: Equality of the Vanishing Subspaces $$VJN_p$$ and $$CJN_p$$

Abstract

AbstractThe John–Nirenberg spaces $$JN_p$$ J N p are generalizations of the space of bounded mean oscillation BMO with $$JN_\infty =BMO$$ J N ∞ = B M O . Their vanishing subspaces $$VJN_p$$ V J N p and $$CJN_p$$ C J N p are defined in similar ways as VMO and CMO, which are subspaces of BMO. As our main result, we prove that $$VJN_p$$ V J N p and $$CJN_p$$ C J 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 ( R n ) = L p ( R n ) / R , if $$p = q$$ p = q .