Nontrivial examples of $JN_p$ and $VJN_p$ functions

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