Dyadic John-Nirenberg space

Kinnunen, Juha; Myyryläinen, Kim

doi:10.48550/arxiv.2107.00492

Cited by 3 publications

(3 citation statements)

References 21 publications

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“…• Berkovits et al [12] applied the dyadic variant of JN p (Q 0 ) in the study of self-improving properties of some Poincaré-type inequalities. Later, the dyadic JN p (Q 0 ) was further studied by Kinnunen and Myyryläinen in [60].…”

Section: John-nirenberg Space Jn Pmentioning

confidence: 99%

A Survey on Function Spaces of John--Nirenberg Type

Tao,

Yang,

Yuan

2021

Preprint

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Section: John-nirenberg Space Jn Pmentioning

confidence: 99%

A Survey on Function Spaces of John--Nirenberg Type

Tao,

Yang,

Yuan

2021

Preprint

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“…Other related function spaces include the dyadic J N p [8], the John-Nirenberg-Campanato spaces [15], their localized versions [14] and the sparse J N p [5]. For an extensive survey of John-Nirenberg type spaces see [17].…”

Section: Introductionmentioning

confidence: 99%

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

Takala

2022

Math. Z.

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We 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.

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“…Other related function spaces include the dyadic JN p [8], the John-Nirenberg-Campanato spaces [15], their localized versions [14] and the sparse JN p [5]. For an extensive survey of John-Nirenberg type spaces see [17].…”

Section: Introductionmentioning

confidence: 99%

Nontrivial examples of $JN_p$ and $VJN_p$ functions

Takala

2021

Preprint

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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, proving the nontriviality of the vanishing subspace V JNp. Finally we extend some results from JNp type functions defined in a bounded cube to functions that are defined in the whole space R n .

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Dyadic John-Nirenberg space

Cited by 3 publications

References 21 publications

A Survey on Function Spaces of John--Nirenberg Type

A Survey on Function Spaces of John--Nirenberg Type

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

Nontrivial examples of $JN_p$ and $VJN_p$ functions

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