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

Abstract

AbstractWe study the John-Nirenberg space $$JN_p$$ J N p , which is a generalization of the space of bounded mean oscillation. In this paper we construct new $$JN_p$$ J N p functions, that increase the understanding of this function space. It is already known that $$L^p(Q_0) \subsetneq JN_p(Q_0) \subsetneq L^{p,\infty }(Q_0)$$ L p ( Q 0 ) ⊊ J N p ( Q 0 ) ⊊ L p , ∞ ( Q 0 ) . We show that if $$|f|^{1/p} \in JN_p(Q_0)$$ | f | 1 / p ∈ J N p ( Q 0 ) , then $$|f|^{1/q} \in JN_q(Q_0)$$ | f | 1 / q ∈ J N q ( Q 0 ) , where $$q \ge p$$ q ≥ p , but there exists a nonnegative function f such that $$f^{1/p} \notin JN_p(Q_0)$$ f 1 / p ∉ J N p ( Q 0 ) even though $$f^{1/q} \in JN_q(Q_0)$$ f 1 / q ∈ J N q ( Q 0 ) , for every $$q \in (p,\infty )$$ q ∈ ( p , ∞ ) . We present functions in $$JN_p(Q_0) \setminus VJN_p(Q_0)$$ J N p ( Q 0 ) \ V J N p ( Q 0 ) and in $$VJN_p(Q_0) {\setminus } L^p(Q_0)$$ V J N p ( Q 0 ) \ L p ( Q 0 ) , proving the nontriviality of the vanishing subspace $$VJN_p$$ V J N p , which is a $$JN_p$$ J N p space version of VMO. We prove the embedding $$JN_p({\mathbb {R}}^n) \subset L^{p,\infty }({\mathbb {R}}^n)/{\mathbb {R}}$$ J N p ( R n ) ⊂ L p , ∞ ( R n ) / R . Finally we show that we can extend the constructed functions into $${\mathbb {R}}^n$$ R n , such that we get a function in $$JN_p({\mathbb {R}}^n) {\setminus } VJN_p({\mathbb {R}}^n)$$ J N p ( R n ) \ V J N p ( R n ) and another in $$CJN_p({\mathbb {R}}^n) {\setminus } L^p({\mathbb {R}}^n)/{\mathbb {R}}$$ C J N p ( R n ) \ L p ( R n ) / R . Here $$CJN_p$$ C J N p is a subspace of $$JN_p$$ J N p that is inspired by the space CMO.