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

Abstract

Let ${\Lambda }_s$ denote the inhomogeneous Lipschitz space of order $s\in (0,\infty )$ on ${\mathbb{R}}^n$. This article characterizes the distance $d(f,V{)}_{{\Lambda }_s}:={\inf }_{g\in V}\Vert f-g{\Vert }_{{\Lambda }_s}$ 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|(-Δ)^{\frac {αr}2} (T_{α, t^α} f)(x) \right|\leq λt^{s -rα}\ \ \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 $0<\epsilon <{\lambda }_{\alpha ,r,s}(f)$, let $$D_{\alpha ,r}(s,f,\epsilon ):=\left\{(x,t)\in {\mathbb{R}}^n\times (0,1]:\left|(-\Delta {)}^{\frac{\alpha r}{2}}(T_{\alpha ,t^{\alpha }}f)(x)\right|>\epsilon t^{s-r\alpha }\right\}$$ 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} ${\epsilon }_{\alpha ,r,s,\nu }(f):=\inf \{\epsilon \in (0,\infty ):\nu (D_{\alpha ,r}(s,f,\epsilon ))<\infty \}.$ 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 ${\epsilon }_{\alpha ,r,s,\nu }(f)\sim \mathrm{dist}(f,V{)}_{{\Lambda }_s}.$

46 pages; Submitted