Nontrivial examples of JNp and VJNp functions

Timo Takala*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

3 Citations (Scopus)
46 Downloads (Pure)

Abstract

We study the John-Nirenberg space JNp, which is a generalization of the space of bounded mean oscillation. In this paper we construct new JNp functions, that increase the understanding of this function space. It is already known that Lp(Q) ⊊ JNp(Q) ⊊ Lp,(Q). We show that if | f| 1/p∈ JNp(Q) , then | f| 1/q∈ JNq(Q) , where q≥ p, but there exists a nonnegative function f such that f1/p∉ JNp(Q) even though f1/q∈ JNq(Q) , for every q∈ (p, ∞). We present functions in JNp(Q) \ VJNp(Q) and in VJNp(Q) \ Lp(Q) , proving the nontriviality of the vanishing subspace VJNp, which is a JNp space version of VMO. We prove the embedding JNp(Rn) ⊂ Lp,(Rn) / R. Finally we show that we can extend the constructed functions into Rn, such that we get a function in JNp(Rn) \ VJNp(Rn) and another in CJNp(Rn) \ Lp(Rn) / R. Here CJNp is a subspace of JNp that is inspired by the space CMO.

Original languageEnglish
Pages (from-to)1279-1305
Number of pages27
JournalMATHEMATISCHE ZEITSCHRIFT
Volume302
Issue number2
DOIs
Publication statusPublished - Oct 2022
MoE publication typeA1 Journal article-refereed

Keywords

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

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