John–Nirenberg lemmas for a doubling measure

Aalto, Daniel; Berkovits, Lauri; Kansanen, Outi Elina; Hong, Yi

doi:10.4064/sm204-1-2

Cited by 41 publications

(118 citation statements)

References 10 publications

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“…One can show that this is a norm on JN p (Ω) modulo constants. These spaces have been considered in the case where Ω is a cube in [19,37] and a general Euclidean domain in [33], and generalised to a metric measure space in [1,52]. While it is well known that L p (Ω) ⊂ JN p (Ω) ⊂ L p,∞ (Ω), the strictness of these inclusions has only recently been addressed ( [1,19]).…”

Section: Rearrangements and The Absolute Valuementioning

confidence: 99%

𝐁𝐌𝐎 on shapes and sharp constants

Dafni¹,

Gibara²

2020

Contemporary Mathematics

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We consider a very general definition of BMO on a domain in R n , where the mean oscillation is taken with respect to a basis of shapes, i.e. a collection of open sets covering the domain. We examine the basic properties and various inequalities that can be proved for such functions, with special emphasis on sharp constants. For the standard bases of shapes consisting of balls or cubes (classic BMO), or rectangles (strong BMO), we review known results, such as the boundedness of rearrangements and its consequences. Finally, we prove a product decomposition for BMO when the shapes exhibit some product structure, as in the case of strong BMO.

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Section: Rearrangements and The Absolute Valuementioning

confidence: 99%

𝐁𝐌𝐎 on shapes and sharp constants

Dafni¹,

Gibara²

2020

Contemporary Mathematics

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“…In the same paper, John and Nirenberg introduced a larger space of functions. As opposed to any BMO function, that has exponentially decaying distribution function, a function in this larger space is known to belong to a weak L p -space, [12,Lemma 3]; the inclusion being strict, see [1,Example 3.5]. We extend this weak-type inequality to the case of John domains.…”

Section: Introductionmentioning

confidence: 99%

“…Then there exist constants σ, τ ∈ N and a chain decomposition {C(Q) : Q ∈ W(G)} of G with the following conditions (1)- (3):…”

Section: Proposition (Chain Decomposition) Suppose 1 < P < ∞ and G Imentioning

confidence: 99%

“…It is shown in [12, Lemma 3] that a function satisfying (1.1), with G being a cube Q in R n , belongs to a weak L p (Q)-space. More precisely, there exists a positive constant C, depending only on n and p, so that for all f ∈ L 1 (Q),We refer to [7,19,1] for other proofs of this result.We mention papers [5,6,15,16] where a related discrete summability condition is studied, and a recent paper [2] where its relation to condition (1.1) is discussed. In [5], in particular, the authors prove a local to global result in connection with this discrete summability condition.…”

mentioning

confidence: 99%

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Aspects of local-to-global results

Hurri-Syrjänen¹,

Marola²,

Vähäkangas³

2014

Bulletin of the London Mathematical Society

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Abstract. We establish local to global results for a function space which is larger than the well known BMO space, and was also introduced by John and Nirenberg. IntroductionThe space of functions of bounded mean oscillation, abbreviated to BMO, is introduced by John and Nirenberg [12]. In the same paper, John and Nirenberg introduced a larger space of functions. As opposed to any BMO function, that has exponentially decaying distribution function, a function in this larger space is known to belong to a weak L p -space, [12, Lemma 3]; the inclusion being strict, see [1, Example 3.5]. We extend this weak-type inequality to the case of John domains. The equivalence of local and global BMO norms is a rather well-known result, due to Reimann and Rychener [17]. We obtain the corresponding local to global result for the mentioned larger space of functions.Let G be a proper open subset of R n , n ≥ 1. The following condition was introduced in [12]: Let f : G → R be a function in L 1 (G) and let us assume that there exists 1 < p < ∞ such thatwhere the supremum is taken over all partitions P(G) of G into cubes such that Q ⊂ G for each Q ∈ P(G), the interiors of these cubes are pairwise disjoint, and G = Q∈P(G) Q. We call such partitions admissible. It is shown in [12, Lemma 3] that a function satisfying (1.1), with G being a cube Q in R n , belongs to a weak L p (Q)-space. More precisely, there exists a positive constant C, depending only on n and p, so that for all f ∈ L 1 (Q),We refer to [7,19,1] for other proofs of this result.We mention papers [5,6,15,16] where a related discrete summability condition is studied, and a recent paper [2] where its relation to condition (1.1) is discussed. In [5], in particular, the authors prove a local to global result in connection with this discrete summability condition.

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“…It is left to reader to decide if the author of the present paper is demonstrating his own obtuseness. 7 The fact that the containment is strict was shown in [1]. 8 See also [12] for related inequalities.…”

Section: We Letmentioning

confidence: 99%

Marcinkiewicz spaces, Garsia-Rodemich spaces and the scale of John-Nirenberg self improving inequalities

Milman¹

2016

Ann. Acad. Sci. Fenn. Math.

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Abstract. We extend to n-dimensions a characterization of the Marcinkiewicz L(p, ∞) spaces first obtained by Garsia-Rodemich in the one dimensional case. This leads to a new proof of the John-Nirenberg self-improving inequalities. We also show a related result that provides still a new characterization of the L(p, ∞) spaces in terms of distribution functions, reflects the self-improving inequalities directly, and also characterizes L(∞, ∞), the rearrangement invariant hull of BM O. We show an application to the study of tensor products with L(∞, ∞) spaces, which complements the classical work of O'Neil [19] and the more recent work of Astashkin [2].

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John–Nirenberg lemmas for a doubling measure

Cited by 41 publications

References 10 publications

𝐁𝐌𝐎 on shapes and sharp constants

𝐁𝐌𝐎 on shapes and sharp constants

Aspects of local-to-global results

Marcinkiewicz spaces, Garsia-Rodemich spaces and the scale of John-Nirenberg self improving inequalities

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