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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 theHardy--Littlewood maximal operator on the associate space of theconvexification of $X$, the authors prove that, for any locally integrablefunction $f$ 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 constantsindependent 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 truefor 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. VanSchaftingen, and P.-L. Yung, and which in other cases are new. Thisgeneralization has a wide range of applications and, particularly, enables theauthors to establish new fractional Sobolev and new Gagliardo--Nirenberginequalities in various function spaces, including Morrey spaces, mixed-normLebesgue spaces, variable Lebesgue spaces, weighted Lebesgue spaces, Orliczspaces, and Orlicz-slice (generalized amalgam) spaces, and, even in all thesespecial cases, the obtained results are new. The proofs of these resultsstrongly depend on the Poincar\'e inequality, the extrapolation, the exactoperator norm on $X'$ of the Hardy--Littlewood maximal operator, and theexquisite geometry of $\mathbb{R}^n.$

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