Nontrivial examples of JNp and VJNp functions

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Journal Title
Journal ISSN
Volume Title
A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
Date
2022-10
Major/Subject
Mcode
Degree programme
Language
en
Pages
27
1279-1305
Series
MATHEMATISCHE ZEITSCHRIFT, Volume 302, issue 2
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.
Description
Funding Information: Open Access funding provided by Aalto University. The research was funded by Aalto University. Publisher Copyright: © 2022, The Author(s).
Keywords
Bounded mean oscillation, Cube, Euclidian space, John-Nirenberg inequality, John–Nirenberg space, Vanishing subspace
Other note
Citation
Takala , T 2022 , ' Nontrivial examples of JN p and VJN p functions ' , MATHEMATISCHE ZEITSCHRIFT , vol. 302 , no. 2 , pp. 1279-1305 . https://doi.org/10.1007/s00209-022-03100-w