Difference and Wavelet Characterizations of Distances from Functions in Lipschitz Spaces to Their Subspaces
Feng Dai, Eero Saksman, Dachun Yang, Wen Yuan, Yangyang Zhang
TL;DR
This work develops a unified, wavelet- and difference-based framework to quantify the distance from functions in Lipschitz spaces $\Lambda_s$ to a broad class of subspaces $V\cap\Lambda_s$, including Sobolev, Besov, and Triebel--Lizorkin-type spaces and their Besov-/Triebel--Lizorkin-type variants. The authors introduce the Daubechies $s$-Lipschitz $X$-based spaces $\Lambda_X^{s}$ and prove that $\operatorname{dist}(f,\Lambda_X^{s})_{\Lambda_s}$ is equivalent to a Lipschitz deviation constant $\varepsilon_X f$ (and, in endpoint cases, to a Carleson-measure–controlled quantity $\varepsilon_{X,\nu}f$). They establish precise wavelet-characterizations of these distances via the tails of wavelet coefficients and derive corresponding closure descriptions, then connect these results to hyperbolic geometry through Carleson measures. Applications to Sobolev, Besov, Triebel--Lizorkin, Besov-type, and Triebel--Lizorkin-type spaces yield new, explicit distance formulas, generalizing prior BMO/Lipschitz results to higher smoothness and a wider array of function spaces. The framework thus provides powerful, transferable tools for distance and approximation problems in harmonic analysis across a broad spectrum of function spaces.
Abstract
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.