Difference and Wavelet Characterizations of Distances from Functions in Lipschitz Spaces to Their Subspaces

Abstract

Let $\hat{I}{›}_s$ 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 $\hat{I}{”}_h^r$ is the usual $r$-th order difference operator with step $h\in {\mathbb{R}}^n$. For each $f\in \hat{I}{›}_s$ and $\epsilon \in (0,L)$, let $S(f,\epsilon ):=\{(x,y)\in {\mathbb{R}}^n\times [0,1]:\frac{\hat{I}{”}_{r}f(x,y)}{y^s}>\epsilon \}$, and let $\hat{I}¼:\mathcal{B}({\mathbb{R}}_{+}^{n+1})\to [0,\infty ]$ be a suitably defined nonnegative extended real-valued function on the Borel $\ddot{I}ƒ$-algebra of subsets of ${\mathbb{R}}_{+}^{n+1}$. Let $\epsilon (f)$ be the infimum of all $\epsilon \in (0,\infty )$ such that $\hat{I}¼(S(f,\epsilon ))<\infty$. The main target of this article is to characterize the distance from $f$ to a subspace $V\cap \hat{I}{›}_s$ of $\hat{I}{›}_s$ for various function spaces $V$ (including Sobolev, Besov--Triebel--Lizorkin, and Besov--Triebel--Lizorkin-type spaces) in terms of $\epsilon (f)$, 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.