## Abstract

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) ⊊ JN_{p}(Q) ⊊ L^{p}^{,}^{∞}(Q). We show that if | f| ^{1}^{/}^{p}∈ JN_{p}(Q) , then | f| ^{1}^{/}^{q}∈ JN_{q}(Q) , where q≥ p, but there exists a nonnegative function f such that f^{1}^{/}^{p}∉ JN_{p}(Q) even though f^{1}^{/}^{q}∈ JN_{q}(Q) , for every q∈ (p, ∞). We present functions in JN_{p}(Q) \ VJN_{p}(Q) and in VJN_{p}(Q) \ L^{p}(Q) , proving the nontriviality of the vanishing subspace VJN_{p}, which is a JN_{p} space version of VMO. We prove the embedding JN_{p}(R^{n}) ⊂ L^{p}^{,}^{∞}(R^{n}) / R. Finally we show that we can extend the constructed functions into R^{n}, such that we get a function in JN_{p}(R^{n}) \ VJN_{p}(R^{n}) and another in CJN_{p}(R^{n}) \ L^{p}(R^{n}) / R. Here CJN_{p} is a subspace of JN_{p} that is inspired by the space CMO.

Original language | English |
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Pages (from-to) | 1279-1305 |

Number of pages | 27 |

Journal | MATHEMATISCHE ZEITSCHRIFT |

Volume | 302 |

Issue number | 2 |

DOIs | |

Publication status | Published - Oct 2022 |

MoE publication type | A1 Journal article-refereed |

## Keywords

- Bounded mean oscillation
- Cube
- Euclidian space
- John-Nirenberg inequality
- John–Nirenberg space
- Vanishing subspace

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