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In this article, via first establishing a weighted variant of the profound and far-reaching inequality obtained by A. Cohen, W. Dahmen, I. Daubechies, and R. DeVore in 2003, the authors give two new characterizations of Muckenhoupt weights. As an application, the authors further establish a representation formula of gradients with sharp parameters in ball Banach function spaces, which extends the famous formula obtained by H. Brezis, A. Seeger, J. Van Schaftingen, and P.-L. Yung in 2021 from classical Sobolev spaces to various different Sobolev-type spaces and gives an affirmative answer to the question in page 29 of [Calc. Var. Partial Differential Equations 62 (2023), Paper No. 234]. The most novelty of this article exists in subtly revealing the mutual equivalences among the Muckenhoupt weight, the weighted variant of the inequality of Cohen et al., and the weighted upper estimate of the formula of Brezis et al., Comment: 48 pages; Submitted

Let $X$ be a ball Banach function space on $\mathbb{R}^n$. In this article, under some mild assumptions about both $X$ and the boundedness of the Hardy--Littlewood maximal operator on both $X$ and the associate space of its convexification, the authors successfully recover the homogeneous ball Banach Sobolev semi-norm $\|\,|\nabla f|\,\|_X$ via the functional $$ \sup_{\lambda\in(0,\infty)}\lambda \left\|\left[\int_{\{y\in\mathbb{R}^n:\ |f(\cdot)-f(y)|>\lambda|\cdot-y|^{1+\frac{\gamma}{q}}\}} \left|\cdot-y\right|^{\gamma-n}\,dy\right]^\frac{1}{q}\right\|_X $$ for any distributions $f$ with $|\nabla f|\in X$, as well as the corresponding limiting identities with the limit for $\lambda\to\infty$ when $\gamma\in(0,\infty)$ or the limit for $\lambda\to0^+$ when $\gamma\in(-\infty,0)$, where $\gamma\in\mathbb{R}\setminus\{0\}$ and where $q\in(0,\infty)$ is related to $X$. In particular, some of these results are still new even when $X:=L^p(X)$ with $p\in[1,\infty)$. As applications, the authors obtain some fractional Sobolev-type and some fractional Gagliardo--Nirenberg-type inequalities in the setting of $X$. All these results are of quite wide generality and are applied to various specific function spaces, including Morrey, mixed-norm (or variable or weighted) Lebesgue, Lorentz, and Orlicz (or Orlicz-slice) spaces, some of which are new even in all these special cases. The novelty of this article is to use both the method of the extrapolation and the boundedness of the Hardy--Littlewood maximal operator on both $X$ and the associate space of its convexification to overcome the essential difficulties caused by the deficiency of both the translation and the rotation invariance and an explicit expression of the norm of $X$., Comment: 80 pages; Submitted

Let $X$ be a ball Banach function space on ${\mathbb R}^n$. In this article, under some mild assumptions about both $X$ and the boundedness of the Hardy--Littlewood maximal operator on the associate space of the convexification of $X$, the authors prove that, for any locally integrable function $f$ with $\|\,|\nabla f|\,\|_{X}<\infty$, $$\sup_{\lambda\in(0,\infty)}\lambda\left \|\left|\left\{y\in{\mathbb R}^n:\ |f(\cdot)-f(y)| >\lambda|\cdot-y|^{\frac{n}{q}+1}\right\}\right|^{\frac{1}{q}} \right\|_X\sim \|\,|\nabla f|\,\|_X$$ with the positive equivalence constants independent of $f$, where the index $q\in(0,\infty)$ is related to $X$ and $|\{y\in{\mathbb R}^n:\ |f(\cdot)-f(y)| >\lambda|\cdot-y|^{\frac{n}{q}+1}\}|$ is the Lebesgue measure of the set under consideration. In particular, when $X:=L^p({\mathbb R}^n)$ with $p\in [1,\infty)$, the above formulae hold true for any given $q\in (0,\infty)$ with $n(\frac{1}{p}-\frac{1}{q})<1$, which when $q=p$ are exactly the recent surprising formulae of H. Brezis, J. Van Schaftingen, and P.-L. Yung, and which in other cases are new. This generalization has a wide range of applications and, particularly, enables the authors to establish new fractional Sobolev and new 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 Poincar\'e inequality, the extrapolation, the exact operator norm on $X'$ of the Hardy--Littlewood maximal operator, and the exquisite geometry of $\mathbb{R}^n.$, Comment: 68 pages, Submitted