Title: Sharp Weighted Cohen--Dahmen--Daubechies--DeVore Inequality with Applications to (Weighted) Critical Sobolev Spaces, Gagliardo--Nirenberg Inequalities, and Muckenhoupt Weights Show Chinese title

Title: 带权的Cohen--Dahmen--Daubechies--DeVore不等式及其在（带权）临界Sobolev空间、Gagliardo--Nirenberg不等式和Muckenhoupt权中的应用

Abstract: 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 $A_p$-weight constant for any $p\in[1,\infty)$ 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 in 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 most novelty of this article exists in applying and further developing the good cube method introduced by Cohen et al.\ to trace the sharp dependences on weight constants. Show Chinese abstract

Abstract: 在本文中，我们建立了一个由A. Cohen、W. Dahmen、I. Daubechies和R. DeVore于2003年获得的深远不等式的定量加权变体，该不等式对于任何$p\in[1,\infty)$的$A_p$-权常数的依赖关系是精确的。 作为应用，我们得到了关于小波的临界加权Sobolev空间的几乎刻画，该加权Sobolev空间与加权Besov空间之间的精确实插值，以及球Banach函数空间框架下的三个新的Gagliardo--Nirenberg型不等式。 此外，我们将这一精确的加权不等式应用于扩展球Banach函数空间中的著名Brezis--Seeger--Van Schaftingen--Yung公式，这为[Calc. Var. Partial Differential Equations 62 (2023), Paper No. 234]第29页的问题提供了肯定回答。 值得注意的是，我们进一步建立了与Cohen等人不等式和Brezis等人公式相关的Muckenhoupt权的两个新刻画。 本文最具创新性之处在于应用并进一步发展了Cohen等人引入的优良立方体方法，以追踪权常数的精确依赖关系。