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

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.