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<p>In this article, we establish a quantitative weighted variant of a far-reaching inequality obtained by A. Cohen, W. Dahmen, I. Daubechies, and R. DeVore in 2003, whose dependence on the <i>A</i><sub><i>p</i></sub>-weight constant for any <i>p</i> ∈ [1, ∞) is sharp. As applications, we obtain the almost characterization of the critical weighted Sobolev space in terms of wavelets, a sharp real interpolation between this weighted Sobolev space and weighted Besov spaces, and three new Gagliardo–Nirenberg type inequalities in the framework of ball Banach function spaces. Moreover, we apply this sharp weighted inequality to extend the famous Brezis–Seeger–Van Schaftingen–Yung formula in ball Banach function spaces, which gives an affirmative answer to the question on Page 29 of [Calc Var Partial Differential Equations 62 (2023), Paper No. 234]. Notably, we further establish two new characterizations of Muckenhoupt weights related to the inequality of Cohen et al. and the formula of Brezis et al. The main novelty of this article exists in applying and further developing the good cube method introduced by Cohen et al. to trace the sharp dependences of positive constants appearing in the inequality under consideration on weight constants.</p>

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Sharp weighted Cohen–Dahmen–Daubechies–DeVore inequality with applications to (weighted) critical Sobolev spaces, Gagliardo–Nirenberg inequalities, and Muckenhoupt weights

  • 期刊论文

  • 发表日期：

    2026年5月6日

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  • Yinqin Li,
  • Dachun Yang,
  • Wen Yuan,
  • Yangyang Zhang,
  • Yirui Zhao

摘要

In this article, we establish a quantitative weighted variant of a far-reaching inequality obtained by A. Cohen, W. Dahmen, I. Daubechies, and R. DeVore in 2003, whose dependence on the Ap-weight constant for any p ∈ [1, ∞) is sharp. As applications, we obtain the almost characterization of the critical weighted Sobolev space in terms of wavelets, a sharp real interpolation between this weighted Sobolev space and weighted Besov spaces, and three new Gagliardo–Nirenberg type inequalities in the framework of ball Banach function spaces. Moreover, we apply this sharp weighted inequality to extend the famous Brezis–Seeger–Van Schaftingen–Yung formula in ball Banach function spaces, which gives an affirmative answer to the question on Page 29 of [Calc Var Partial Differential Equations 62 (2023), Paper No. 234]. Notably, we further establish two new characterizations of Muckenhoupt weights related to the inequality of Cohen et al. and the formula of Brezis et al. The main novelty of this article exists in applying and further developing the good cube method introduced by Cohen et al. to trace the sharp dependences of positive constants appearing in the inequality under consideration on weight constants.