Nontrivial examples of $$JN_p$$ and $$VJN_p$$ functions

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