Let X be a ball Banach function space on \({\mathbb {R}}^n\) , \(k\in {\mathbb {N}}\) , \(h\in {\mathbb {R}}^n\) , and \(\Delta ^k_h\) denote the k-th order difference. In this article, under some mild additional assumptions about X, the authors prove that, for both parameters q and \(\gamma \) in sharp ranges which are related to X and for any locally integrable function f on \({{\mathbb {R}}^n}\) satisfying \(|\nabla ^k f|\in X\) , \(\begin{aligned} \sup _{\lambda \in (0,\infty )}\lambda \left\| \left[ \int _{\{h\in {\mathbb {R}}^n:\ |\Delta _h^k f(\cdot )|>\lambda |h|^{k+\frac{\gamma }{q}}\}} \left| h\right| ^{\gamma -n}\,dh\right] ^\frac{1}{q}\right\| _X \sim \left\| \,\left| \nabla ^k f\right| \,\right\| _{X} \end{aligned}\) with the positive equivalence constants independent of f. As applications, the authors establish the Brezis–Seeger–Van Schaftingen–Yung (for short, BSVY) characterization of higher-order homogeneous ball Banach Sobolev spaces and higher-order fractional Gagliardo–Nirenberg and Sobolev type inequalities in critical cases. All these results are of quite wide generality and can be applied to various specific function spaces; moreover, even when \(X:= L^{q}\) , these results when \(k=1\) coincide with the best known results and when \(k\ge 2\) are completely new. The first novelty is to establish a sparse characterization of dyadic cubes in level sets related to the higher-order local approximation, which, together with the well-known Whitney inequality in approximation theory, further induces a higher-order weighted variant of the remarkable inequality obtained by A. Cohen, W. Dahmen, I. Daubechies, and R. DeVore; the second novelty is to combine this weighted inequality neatly with a variant higher-order Poincaré inequality to establish the desired upper estimate of BSVY formulae in weighted Lebesgue spaces.
