The John–Nirenberg Space: Equality of the Vanishing Subspaces VJNp and CJNp

Riikka Korte; Timo Takala

doi:10.1007/s12220-023-01512-6

The John–Nirenberg Space: Equality of the Vanishing Subspaces VJN_p and CJN_p

Riikka Korte, Timo Takala^*

^*Corresponding author for this work

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Abstract

The John–Nirenberg spaces JN_p are generalizations of the space of bounded mean oscillation BMO with JN_∞= BMO . Their vanishing subspaces VJN_p and CJN_p are defined in similar ways as VMO and CMO, which are subspaces of BMO. As our main result, we prove that VJN_p and CJN_p coincide by showing that certain Morrey type integrals of JN_p functions tend to zero for small and large cubes. We also show that JN_p_,_q(Rⁿ) = L^p(Rⁿ) / R , if p= q .

Original language	English
Article number	67
Pages (from-to)	1-19
Number of pages	19
Journal	Journal of Geometric Analysis
Volume	34
Issue number	3
DOIs	https://doi.org/10.1007/s12220-023-01512-6
Publication status	Published - Mar 2024
MoE publication type	A1 Journal article-refereed

Keywords

Bounded mean oscillation
Euclidean space
John–Nirenberg inequality
John–Nirenberg space
Morrey type integral
Vanishing subspace

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title = "The John–Nirenberg Space: Equality of the Vanishing Subspaces VJNp and CJNp",

abstract = "The John–Nirenberg spaces JNp are generalizations of the space of bounded mean oscillation BMO with JN∞= BMO . Their vanishing subspaces VJNp and CJNp are defined in similar ways as VMO and CMO, which are subspaces of BMO. As our main result, we prove that VJNp and CJNp coincide by showing that certain Morrey type integrals of JNp functions tend to zero for small and large cubes. We also show that JNp,q(Rn) = Lp(Rn) / R , if p= q .",

keywords = "Bounded mean oscillation, Euclidean space, John–Nirenberg inequality, John–Nirenberg space, Morrey type integral, Vanishing subspace",

author = "Riikka Korte and Timo Takala",

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language = "English",

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T1 - The John–Nirenberg Space

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AU - Korte, Riikka

AU - Takala, Timo

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N2 - The John–Nirenberg spaces JNp are generalizations of the space of bounded mean oscillation BMO with JN∞= BMO . Their vanishing subspaces VJNp and CJNp are defined in similar ways as VMO and CMO, which are subspaces of BMO. As our main result, we prove that VJNp and CJNp coincide by showing that certain Morrey type integrals of JNp functions tend to zero for small and large cubes. We also show that JNp,q(Rn) = Lp(Rn) / R , if p= q .

AB - The John–Nirenberg spaces JNp are generalizations of the space of bounded mean oscillation BMO with JN∞= BMO . Their vanishing subspaces VJNp and CJNp are defined in similar ways as VMO and CMO, which are subspaces of BMO. As our main result, we prove that VJNp and CJNp coincide by showing that certain Morrey type integrals of JNp functions tend to zero for small and large cubes. We also show that JNp,q(Rn) = Lp(Rn) / R , if p= q .

KW - Bounded mean oscillation

KW - Euclidean space

KW - John–Nirenberg inequality

KW - John–Nirenberg space

KW - Morrey type integral

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The John–Nirenberg Space: Equality of the Vanishing Subspaces VJN_p and CJN_p

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