Median-Type John–Nirenberg Space in Metric Measure Spaces

Myyryläinen, Kim

doi:10.1007/s12220-022-00872-9

Cited by 8 publications

(5 citation statements)

References 32 publications

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“…In the next lemma, we list the basic properties of the maximal s-median. We refer to [20,21] for the proofs of the properties. Lemma 6.2 Let 0 < s ≤ 1.…”

Section: Declarationsmentioning

confidence: 99%

John–Nirenberg inequalities for parabolic BMO

2022

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We discuss a parabolic version of the space of functions of bounded mean oscillation related to a doubly nonlinear parabolic partial differential equation. Parabolic John–Nirenberg inequalities, which give exponential decay estimates for the oscillation of a function, are shown in the natural geometry of the partial differential equation. Chaining arguments are applied to change the time lag in the parabolic John–Nirenberg inequality. We also show that the quasihyperbolic boundary condition is a necessary and sufficient condition for a global parabolic John–Nirenberg inequality. Moreover, we consider John–Nirenberg inequalities with medians instead of integral averages and show that this approach gives the same class of functions as the original definition.

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“…In the next lemma, we list the basic properties of the maximal s-median. We refer to [20,21] for the proofs of the properties. Lemma 6.2 Let 0 < s ≤ 1.…”

Section: Declarationsmentioning

confidence: 99%

John–Nirenberg inequalities for parabolic BMO

2022

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“…One can also define J N p by using medians which allows us to not use integrals at all. This approach is studied by Myyryläinen [11]. We do not consider J N p with p = 1, because clearly…”

Section: Preliminariesmentioning

confidence: 99%

“…This is because the space depends on how much we let the sets Q i overlap. For example many of the definitions in the more general metric measure space, such as in [1,6,9,11], use balls B i instead of cubes, and these balls may overlap with each other in some definitions. This results in different spaces.…”

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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“…In the next lemma, we list the basic properties of the maximal s-median. We refer to [20,21] for the proofs of the properties. Lemma 6.2.…”

Section: Parabolic Bmo With Mediansmentioning

confidence: 99%

John-Nirenberg inequalities for parabolic BMO

Kinnunen¹,

Myyryläinen²,

Yang³

2022

Preprint

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We discuss a parabolic version of the space of functions of bounded mean oscillation related to a doubly nonlinear parabolic partial differential equation. Parabolic John-Nirenberg inequalities, which give exponential decay estimates for the oscillation of a function, are shown in the natural geometry of the partial differential equation. Chaining arguments are applied to change the time lag in the parabolic John-Nirenberg inequality. We also show that the quasihyperbolic boundary condition is a necessary and sufficient condition for a global parabolic John-Nirenberg inequality. Moreover, we consider John-Nirenberg inequalities with medians instead of integral averages and show that this approach gives the same class of functions as the original definition. 2020 Mathematics Subject Classification. 42B35, 42B37.

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Median-Type John–Nirenberg Space in Metric Measure Spaces

Cited by 8 publications

References 32 publications

John–Nirenberg inequalities for parabolic BMO

John–Nirenberg inequalities for parabolic BMO

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

John-Nirenberg inequalities for parabolic BMO

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