Difference and Wavelet Characterizations of Distances from Functions in Lipschitz Spaces to Their Subspaces
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_ | 1866916751192621056 |
|---|---|
| author | Dai, Feng Saksman, Eero Yang, Dachun Yuan, Wen Zhang, Yangyang |
| author_facet | Dai, Feng Saksman, Eero Yang, Dachun Yuan, Wen Zhang, Yangyang |
| contents | Let $Λ_s$ denote the Lipschitz space of order $s\in(0,\infty)$ on $\mathbb{R}^n$, which consists of all $f\in\mathfrak{C}\cap L^\infty$ such that, for some constant $L\in(0,\infty)$ and some integer $r\in(s,\infty)$, \begin{equation*} \label{0-1}Δ_r f(x,y): =\sup_{|h|\leq y} |Δ_h^r f(x)|\leq L y^s, \ x\in\mathbb{R}^n, \ y \in(0, 1]. \end{equation*} Here (and throughout the article) $\mathfrak{C}$ refers to continuous functions, and $Δ_h^r$ is the usual $r$-th order difference operator with step $h\in\mathbb{R}^n$. For each $f\in Λ_s$ and $\varepsilon\in(0,L)$, let $ S(f,\varepsilon):= \{ (x,y)\in\mathbb{R}^n\times [0,1]: \frac {Δ_r f(x,y)}{y^s}>\varepsilon\}$, and let $μ: \mathcal{B}(\mathbb{R}_+^{n+1})\to [0,\infty]$ be a suitably defined nonnegative extended real-valued function on the Borel $σ$-algebra of subsets of $\mathbb{R}_+^{n+1}$. Let $\varepsilon(f)$ be the infimum of all $\varepsilon\in(0,\infty)$ such that $μ(S(f,\varepsilon))<\infty$. The main target of this article is to characterize the distance from $f$ to a subspace $V\cap Λ_s$ of $Λ_s$ for various function spaces $V$ (including Sobolev, Besov--Triebel--Lizorkin, and Besov--Triebel--Lizorkin-type spaces) in terms of $\varepsilon(f)$, showing that \begin{equation*} \varepsilon(f)\sim \mathrm{dist} (f, V\cap Λ_s)_{Λ_s}: = \inf_{g\in Λ_s\cap V} \|f-g\|_{Λ_s}.\end{equation*} Moreover, we present our results in a general framework based on quasi-normed lattices of function sequences $X$ and Daubechies $s$-Lipschitz $X$-based spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_16116 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Difference and Wavelet Characterizations of Distances from Functions in Lipschitz Spaces to Their Subspaces Dai, Feng Saksman, Eero Yang, Dachun Yuan, Wen Zhang, Yangyang Functional Analysis Analysis of PDEs Classical Analysis and ODEs Primary 42B35, Secondary 26A16, 42B25, 42C40, 46E35 Let $Λ_s$ denote the Lipschitz space of order $s\in(0,\infty)$ on $\mathbb{R}^n$, which consists of all $f\in\mathfrak{C}\cap L^\infty$ such that, for some constant $L\in(0,\infty)$ and some integer $r\in(s,\infty)$, \begin{equation*} \label{0-1}Δ_r f(x,y): =\sup_{|h|\leq y} |Δ_h^r f(x)|\leq L y^s, \ x\in\mathbb{R}^n, \ y \in(0, 1]. \end{equation*} Here (and throughout the article) $\mathfrak{C}$ refers to continuous functions, and $Δ_h^r$ is the usual $r$-th order difference operator with step $h\in\mathbb{R}^n$. For each $f\in Λ_s$ and $\varepsilon\in(0,L)$, let $ S(f,\varepsilon):= \{ (x,y)\in\mathbb{R}^n\times [0,1]: \frac {Δ_r f(x,y)}{y^s}>\varepsilon\}$, and let $μ: \mathcal{B}(\mathbb{R}_+^{n+1})\to [0,\infty]$ be a suitably defined nonnegative extended real-valued function on the Borel $σ$-algebra of subsets of $\mathbb{R}_+^{n+1}$. Let $\varepsilon(f)$ be the infimum of all $\varepsilon\in(0,\infty)$ such that $μ(S(f,\varepsilon))<\infty$. The main target of this article is to characterize the distance from $f$ to a subspace $V\cap Λ_s$ of $Λ_s$ for various function spaces $V$ (including Sobolev, Besov--Triebel--Lizorkin, and Besov--Triebel--Lizorkin-type spaces) in terms of $\varepsilon(f)$, showing that \begin{equation*} \varepsilon(f)\sim \mathrm{dist} (f, V\cap Λ_s)_{Λ_s}: = \inf_{g\in Λ_s\cap V} \|f-g\|_{Λ_s}.\end{equation*} Moreover, we present our results in a general framework based on quasi-normed lattices of function sequences $X$ and Daubechies $s$-Lipschitz $X$-based spaces. |
| title | Difference and Wavelet Characterizations of Distances from Functions in Lipschitz Spaces to Their Subspaces |
| topic | Functional Analysis Analysis of PDEs Classical Analysis and ODEs Primary 42B35, Secondary 26A16, 42B25, 42C40, 46E35 |
| url | https://arxiv.org/abs/2505.16116 |