Sharp Brezis--Seeger--Van Schaftingen--Yung Formulae for Higher-Order Gradients in Ball Banach Function Spaces
Fuente:
arXiv
Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913851950235648 |
|---|---|
| author | Hu, Pingxu Li, Yinqin Yang, Dachun Yuan, Wen Zhang, Yangyang |
| author_facet | Hu, Pingxu Li, Yinqin Yang, Dachun Yuan, Wen Zhang, Yangyang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_16110 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp Brezis--Seeger--Van Schaftingen--Yung Formulae for Higher-Order Gradients in Ball Banach Function Spaces Hu, Pingxu Li, Yinqin Yang, Dachun Yuan, Wen Zhang, Yangyang Functional Analysis Analysis of PDEs Classical Analysis and ODEs Primary 46E35, Secondary 26D10, 35A23, 42B25, 42B35 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. |
| title | Sharp Brezis--Seeger--Van Schaftingen--Yung Formulae for Higher-Order Gradients in Ball Banach Function Spaces |
| topic | Functional Analysis Analysis of PDEs Classical Analysis and ODEs Primary 46E35, Secondary 26D10, 35A23, 42B25, 42B35 |
| url | https://arxiv.org/abs/2505.16110 |