Sharp Brezis--Seeger--Van Schaftingen--Yung Formulae for Higher-Order Gradients in Ball Banach Function Spaces

TL;DR

This work extends the Brezis–Seeger–Van Schaftingen–Yung (BSVY) formula to higher-order gradients in the general setting of ball Banach function spaces. By developing a sparse-dyadic framework for higher-order local approximation, coupling it with Whitney-type inequalities and a higher-order Poincaré inequality, the authors prove sharp-range equivalences between level-set based quantities involving $|\Delta_h^k f|$ and the $k$-th order gradient norm $\|\nabla^k f\|_X$, under mild structural assumptions on $X$. The results yield a BSVY characterization of $\dot{W}^{k,X}$ and lead to higher-order fractional Gagliardo–Nirenberg and Sobolev inequalities in critical regimes, with robust extrapolation to weighted, mixed-norm, variable, Lorentz, and Orlicz-type spaces. These findings unify and extend known $L^p$-based results, provide new insights for $k\ge 2$ in broad function-space contexts, and supply sharp tools for analyzing higher-order regularity in diverse spaces, including sharp weight characterizations in 1D. Overall, the paper offers a versatile, widely applicable framework for higher-order Sobolev-type analysis in ball Banach spaces, with potential impact on PDE regularity and functional-analytic methods in anisotropic and weighted settings.

Abstract

Let $X$ be a ball Banach function space on $\mathbb{R}^n$, $k\in\mathbb{N}$, $h\in\mathbb{R}^n$, and $Δ^k_h$ denote the $k${\rm th} order difference. In this article, under some mild extra assumptions about $X$, the authors prove that, for both parameters $q$ and $γ$ in \emph{sharp} ranges which are related to $X$ and for any locally integrable function $f$ on ${\mathbb{R}^n}$ satisfying $|\nabla^k f|\in X$, $$ \sup_{λ\in(0,\infty)}λ\left\|\left[\int_{\{h\in\mathbb{R}^n:\ |Δ_h^k f(\cdot)|>λ|h|^{k+\fracγ{q}}\}} \left|h\right|^{γ-n}\,dh\right]^\frac{1}{q}\right\|_X \sim \left\|\,\left|\nabla^k f\right|\,\right\|_{X} $$ 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.

Theorems & Definitions (103)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Definition 2.4