Let \(\Lambda _s\) 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 \(r\in (s,\infty )\) , \(\begin{aligned} \Delta _r f(x,y): =\sup _{|h|\le y} |\Delta _h^r f(x)|\le L y^s, \ x\in \mathbb {R}^n, \ y \in (0, 1], \end{aligned}\) where \(\mathfrak {C}\) refers to continuous functions and \(\Delta _h^r\) is the usual r-th order difference operator with step \(h\in \mathbb {R}^n\) . For each \(f\in \Lambda _s\) and \(\varepsilon \in (0,L)\) , let \( S(f,\varepsilon ):= \left\{ (x,y)\in \mathbb {R}^n\times [0,1]: \frac{\Delta _r f(x,y)}{y^s}>\varepsilon \right\} ,\) and let \(\mu : \mathcal {B}(\mathbb {R}_+^{n+1})\rightarrow [0,\infty ]\) be a suitably defined nonnegative extended real-valued function on the Borel \(\sigma \) -algebra of subsets of \(\mathbb {R}_+^{n+1}\) . Let \(\varepsilon (f)\) be the infimum of all \(\varepsilon \in (0,\infty )\) such that \(\mu (S(f,\varepsilon ))<\infty \) . The main target of this article is to characterize the distance from f to a subspace \(V\cap \Lambda _s\) of \(\Lambda _s\) for various function spaces V (including Sobolev, Besov–Triebel–Lizorkin, and Besov–Triebel–Lizorkin-type spaces) in terms of \(\varepsilon (f)\) , showing that \(\begin{aligned} \varepsilon (f)\sim \text {dist}\, (f, V\cap \Lambda _s)_{\Lambda _s}: = \inf _{g\in \Lambda _s\cap V} \Vert f-g\Vert _{\Lambda _s}.\end{aligned}\) These results extend the characterization of the least constant in the John-Nirenberg inequality via the distance in BMO to \(L^\infty (\mathbb {R}^n)\) by Garnett and Jones in [Ann. of Math. (2) 108 (1978)] as well as the recent work of Saksman and Soler i Gibert in [Canad. J. Math. 74 (2022)] on the distance in \(\Lambda _s\) to the subspace \(\text {J}_s\,(\mathop \text {bmo})\) for any \(s \in (0, 1]\) . 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.
