Fractional Heat Semigroup Characterization of Distances from Functions in Lipschitz Spaces to Their Subspaces

Let $\Lambda_s$ denote the inhomogeneous Lipschitz space of order $s\in(0,\infty)$ on $\mathbb{R}^n$. This article characterizes the distance $d(f, V)_{\Lambda_s}: = \inf_{g\in V} \|f-g\|_{\Lambda_s}$ from a function $f\in \Lambda_s$ to a non-dense subspace $V\subset \Lambda_s$ via the fractional semigroup $\{T_{\alpha, t}: =e^{-t (-\Delta)^{\alpha/2}}: t\in (0, \infty)\}$ for any $\alpha\in(0,\infty)$. Given an integer $r >s/\alpha$, a uniformly bounded continuous function $f$ on $\mathbb{R}^n$ belongs to the space $\Lambda_s$ if and only if there exists a constant $\lambda\in(0,\infty)$ such that \begin{align*} \left|(-\Delta)^{\frac {\alpha r}2} (T_{\alpha, t^\alpha } f)(x) \right|\leq \lambda t^{s -r\alpha }\ \ \text{for any $x\in\mathbb{R}^n$ and $t\in (0, 1]$}.\end{align*} The least such constant is denoted by $\lambda_{ \alpha, r, s}(f)$. For each $f\in \Lambda_s$ and $0<\varepsilon< \lambda_{\alpha,r, s}(f)$, let $$D_{\alpha, r}(s,f,\varepsilon):=\left\{ (x,t)\in \mathbb{R}^n\times (0,1]:\ \left| (-\Delta)^{\frac {\alpha r}2} (T_{\alpha, t^\alpha} f)(x) \right|> \varepsilon t^{s -r \alpha }\right\}$$ be the set of ``bad'' points. To quantify its size, we introduce a class of extended nonnegative \emph{admissible set functions} $\nu$ on the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^n\times [0, 1])$ and define, for any admissible function $\nu$, the \emph{critical index} $\varepsilon_{\alpha, r, s,\nu}(f):=\inf\{\varepsilon\in(0,\infty):\ \nu(D_{\alpha, r}(s,f,\varepsilon))<\infty\}.$ Our result shows that, for a broad class of subspaces $V\subset \Lambda_s$, including intersections of $\Lambda_s$ with Sobolev, Besov, Triebel--Lizorkin, and Besov-type spaces, there exists an admissible function $\nu$ depending on $V$ such that $\varepsilon_{\alpha, r, s,\nu}(f)\sim \mathrm{dist}(f, V)_{\Lambda_s}.$