Nontrivial examples of $J{N}_p$ and $VJ{N}_p$ functions

Abstract

We study the John-Nirenberg space $J{N}_p$, which is a generalization of the space of bounded mean oscillation. In this paper we construct new $J{N}_p$ functions, that increase the understanding of this function space. It is already known that $L^p(Q_0)⊊J{N}_p(Q_0)⊊L^{p,\infty }(Q_0)$. We show that if $|f{|}^{1/p}\in J{N}_p(Q_0)$, then $|f{|}^{1/q}\in J{N}_q(Q_0)$, where $q\ge p$, but there exists a nonnegative function $f$ such that $f^{1/p}\notin J{N}_p(Q_0)$ even though $f^{1/q}\in J{N}_q(Q_0)$, for every $q\in (p,\infty )$. We present functions in $J{N}_p(Q_0)\setminus VJ{N}_p(Q_0)$ and in $VJ{N}_p(Q_0)\setminus L^p(Q_0)$, proving the nontriviality of the vanishing subspace $VJ{N}_p$, which is a $J{N}_p$ space version of $VMO$. We prove the embedding $J{N}_p({\mathbb{R}}^n)\subset L^{p,\infty }({\mathbb{R}}^n)/\mathbb{R}$. Finally we show that we can extend the constructed functions into ${\mathbb{R}}^n$, such that we get a function in $J{N}_p({\mathbb{R}}^n)\setminus VJ{N}_p({\mathbb{R}}^n)$ and another in $CJ{N}_p({\mathbb{R}}^n)\setminus L^p({\mathbb{R}}^n)/\mathbb{R}$. Here $CJ{N}_p$ is a subspace of $J{N}_p$ that is inspired by the space $CMO$.

25 pages

References in corpus (3)

• A Survey on Function Spaces of John--Nirenberg Type
• Median-type John-Nirenberg space in metric measure spaces
• Dyadic John-Nirenberg space

Cited by in corpus (1)

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