Nontrivial examples of JNp and VJNp 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)) subset of J N (p)(Q(0)) subset of L-p(,infinity)(Q(0)). We show that if vertical bar f vertical bar(1/)(p) is an element of J N (p)(Q(0)), then vertical bar f vertical bar(1/)q is an element of J N (q)(Q(0)), where q >= p, but there exists a nonnegative function f such that f(1/p) is not an element of J N-p(Q(0)) even though f(1/q) is an element of J N- p(Q(0)), for every q is an element of (p, infinity). We present functions in J N- p (Q(0)) \ V J N- p(Q(0)) and in V J N- p(Q(0))\L-p(Q(0)), proving the nontriviality of the vanishing subspace V J N- p, which is a J N- p space version of V M O. We prove the embedding J N- p(R-n) subset of L-p,L-infinity(R-n)/R. Finally we show that we can extend the constructed functions into R-n, such that we get a function in J N- p(R-n) \V J N- p(R-n) and another in C J N- p(R-n)\L-p(R-n)/R. Here C J N- p is a subspace of J N- p that is inspired by the space CMO.

Automated summary

This paper studies the John-Nirenberg space JN(p) and constructs new JN(p) functions to improve the understanding of this function space. The nontriviality of the vanishing subspace V JN- p is also proven, providing further insight into this space.