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

Abstract

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 $k${\rm th} order difference. In this article, under some mild extra assumptions about $X$, the authors prove that, for both parameters $q$ and $\gamma$ in \emph{sharp} ranges which are related to $X$ and for any locally integrable function $f$ on ${\mathbb{R}^n}$ satisfying $|\nabla^k f|\in X$, $$ \sup_{\lambda\in(0,\infty)}\lambda \left\|\left[\int_{\{h\in\mathbb{R}^n:\ |\Delta_h^k f(\cdot)|>\lambda|h|^{k+\frac{\gamma}{q}}\}} \left|h\right|^{\gamma-n}\,dh\right]^\frac{1}{q}\right\|_X \sim \left\|\,\left|\nabla^k f\right|\,\right\|_{X} $$ 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 Poincar\'e inequality to establish the desired upper estimate of BSVY formulae in weighted Lebesgue spaces.