A surprising formula for Sobolev norms

Brezis, Haı̈m; Schaftingen, Jean Van; Yung, Po-Lam

doi:10.1073/pnas.2025254118

Cited by 50 publications

(75 citation statements)

References 16 publications

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“…When s ∈ (0, 1), • W s,p (R n ) is usually called the Gagliardo semi-norm. It is well known that, for any s ∈ [1, ∞), W s,p (R n ) contains only the functions that are almost everywhere constant; see, for instance, [8,9] and their references. Also, the same triviality holds true for s ∈ (−∞, 0].…”

Section: Relations With (Fractional) Sobolev Spacesmentioning

confidence: 99%

John--Nirenberg-$Q$ Spaces via Congruent Cubes

Jin¹,

Yang²,

Wen³

2021

Preprint

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To shed some light on the John-Nirenberg space, the authors in this article introduce the John-Nirenberg-Q space via congruent cubes, JNQ α p,q (R n ), which when p = ∞ and q = 2 coincides with the space Q α (R n ) introduced by Essén et al. in [Indiana Univ. Math. J. 49 (2000), 575-615]. Moreover, the authors show that, for some particular indices, JNQ α p,q (R n ) coincides with the congruent John-Nirenberg space, or that the (fractional) Sobolev space is continuously embedded into JNQ α p,q (R n ). Furthermore, the authors characterize JNQ α p,q (R n ) via mean oscillations, and then use this characterization to study the dyadic counterparts. Also, the authors obtain some properties of composition operators on such spaces. The main novelties of this article are twofold: establishing a general equivalence principle for a kind of 'almost increasing' set function introduced in this article, and using the fine geometrical properties of dyadic cubes to properly classify any collection of cubes with pairwise disjoint interiors and equal edge length.

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Section: Relations With (Fractional) Sobolev Spacesmentioning

confidence: 99%

John--Nirenberg-$Q$ Spaces via Congruent Cubes

Jin¹,

Yang²,

Wen³

2021

Preprint

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“…Recently, Brezis, Van Schaftingen and Yung [7] provided a different approach by replacing the strong L p norm in the Gagliardo s-seminorm by the weak L p quasinorm, which complements the BBM formula. Precisely, they proved that there exist a positive constant…”

Section: Introductionmentioning

confidence: 99%

“…Inspired by the technique developed in [7], the proof of Theorem 1.1(a) can be divided by the following two lemmas. The second inequality of (1.14) follows from Lemma 2.3.…”

Section: Introductionmentioning

confidence: 99%

Anisotropic versions of the Brezis-Van Schaftingen-Yung approach at $s=1$ and $s=0$

Gu¹,

Huang²

2022

Preprint

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In 2014, Ludwig showed the limiting behavior of the anisotropic Gagliardo s-seminorm of a function f as s → 1 − and s → 0 + , which extend the results due to Bourgain-Brezis-Mironescu(BBM) and Maz'ya-Shaposhnikova(MS) respectively. Recently, Brezis, Van Schaftingen and Yung provided a different approach by replacing the strong L p norm in the Gagliardo s-seminorm by the weak L p quasinorm. They characterized the case for s = 1 that complements the BBM formula. The corresponding MS formula for s = 0 was later established by Yung and the first author. In this paper, we follow the approach of Brezis-Van Schaftingen-Yung and show the anisotropic versions of s = 1 and s = 0. Our result generalizes the work by Brezis, Van Schaftingen, Yung and the first author and complements the work by Ludwig.

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“…where C (p,n) is a positive constant depending only on p and n. Very recently, Brezis et al [13] discovered an alternative way to repair this defect by replacing the L p norm in (1.1) with the weak L p quasi-norm, namely, • L p,∞ (R n ×R n ) . For any given p ∈ [1, ∞), Brezis et al in [13] proved that there exist positive constants C 1 and C 2 such that, for any…”

Section: Introductionmentioning

confidence: 99%

Generalization in Ball Banach Function Spaces of Brezis--Van Schaftingen--Yung Formulae with Applications to Fractional Sobolev and Gagliardo--Nirenberg Inequalities

Ding¹,

Lin²,

Yang³

et al. 2021

Preprint

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Let X be a ball Banach function space on R n . In this article, under the mild assumption that the Hardy-Littlewood maximal operator is bounded on the associated space X ′ of X, the authors prove that, for anywith the positive equivalence constants independent of f , where q ∈ (0, ∞) is an index depending on the space X, and |E| denotes the Lebesgue measure of a measurable set E ⊂ R n . Particularly, when ), the above estimate holds true for any given q ∈ [1, p], which when q = p is exactly the recent surprising formula of H. Brezis, J. Van Schaftingen, and P.-L. Yung, and which even when q < p is new. This generalization has a wide range of applications and, particularly, enables the authors to establish new fractional Sobolev and Gagliardo-Nirenberg inequalities in various function spaces, including Morrey spaces, mixed-norm Lebesgue spaces, variable Lebesgue spaces, weighted Lebesgue spaces, Orlicz spaces, and Orlicz-slice (generalized amalgam) spaces, and, even in all these special cases, the obtained results are new. The proofs of these results strongly depend on the Poincaré inequality, the extrapolation, the exact operator norm on X ′ of the Hardy-Littlewood maximal operator, and the geometry of R n .

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A surprising formula for Sobolev norms

Cited by 50 publications

References 16 publications

John--Nirenberg-$Q$ Spaces via Congruent Cubes

John--Nirenberg-$Q$ Spaces via Congruent Cubes

Anisotropic versions of the Brezis-Van Schaftingen-Yung approach at $s=1$ and $s=0$

Generalization in Ball Banach Function Spaces of Brezis--Van Schaftingen--Yung Formulae with Applications to Fractional Sobolev and Gagliardo--Nirenberg Inequalities

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