Fractional Heat Semigroup Characterization of Distances from Functions in Lipschitz Spaces to Their Subspaces
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2025
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| _version_ | 1866911129647710208 |
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| author | Dai, Feng Saksman, Eero Yang, Dachun Yuan, Wen Zhang, Yangyang |
| author_facet | Dai, Feng Saksman, Eero Yang, Dachun Yuan, Wen Zhang, Yangyang |
| contents | Let $Λ_s$ denote the inhomogeneous Lipschitz space of order $s\in(0,\infty)$ on $\mathbb{R}^n$. This article characterizes the distance $d(f, V)_{Λ_s}: = \inf_{g\in V} \|f-g\|_{Λ_s}$ from a function $f\in Λ_s$ to a non-dense subspace $V\subset Λ_s$ via the fractional semigroup $\{T_{α, t}: =e^{-t (-Δ)^{α/2}}: t\in (0, \infty)\}$ for any $α\in(0,\infty)$. Given an integer $ r >s/α$, a uniformly bounded continuous function $f$ on $\mathbb{R}^n$ belongs to the space $Λ_s$ if and only if there exists a constant $λ\in(0,\infty)$ 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 $λ_{ α, r, s}(f)$. For each $f\in Λ_s$ and $0<\varepsilon< λ_{α,r, s}(f)$, let $$ D_{α, r}(s,f,\varepsilon):=\left\{ (x,t)\in \mathbb{R}^n\times (0,1]:\ \left| (-Δ)^{\frac {αr}2} (T_{α, t^α} f)(x) \right|> \varepsilon t^{s -r α}\right\}$$ be the set of ``bad'' points. To quantify its size, we introduce a class of extended nonnegative \emph{admissible set functions} $ν$ on the Borel $σ$-algebra $\mathcal{B}(\mathbb{R}^n\times [0, 1])$ and define, for any admissible function $ν$, the \emph{critical index} $ \varepsilon_{α, r, s,ν}(f):=\inf\{\varepsilon\in(0,\infty):\ ν(D_{α, r}(s,f,\varepsilon))<\infty\}.$ Our result shows that, for a broad class of subspaces $V\subset Λ_s$, including intersections of $Λ_s$ with Sobolev, Besov, Triebel--Lizorkin, and Besov-type spaces, there exists an admissible function $ν$ depending on $V$ such that $\varepsilon_{α, r, s,ν}(f)\sim \mathrm{dist}(f, V)_{Λ_s}.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_21269 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fractional Heat Semigroup Characterization of Distances from Functions in Lipschitz Spaces to Their Subspaces Dai, Feng Saksman, Eero Yang, Dachun Yuan, Wen Zhang, Yangyang Functional Analysis Analysis of PDEs Classical Analysis and ODEs Primary 46E35, Secondary 26A16, 35K08, 42C40, 42E35 Let $Λ_s$ denote the inhomogeneous Lipschitz space of order $s\in(0,\infty)$ on $\mathbb{R}^n$. This article characterizes the distance $d(f, V)_{Λ_s}: = \inf_{g\in V} \|f-g\|_{Λ_s}$ from a function $f\in Λ_s$ to a non-dense subspace $V\subset Λ_s$ via the fractional semigroup $\{T_{α, t}: =e^{-t (-Δ)^{α/2}}: t\in (0, \infty)\}$ for any $α\in(0,\infty)$. Given an integer $ r >s/α$, a uniformly bounded continuous function $f$ on $\mathbb{R}^n$ belongs to the space $Λ_s$ if and only if there exists a constant $λ\in(0,\infty)$ 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 $λ_{ α, r, s}(f)$. For each $f\in Λ_s$ and $0<\varepsilon< λ_{α,r, s}(f)$, let $$ D_{α, r}(s,f,\varepsilon):=\left\{ (x,t)\in \mathbb{R}^n\times (0,1]:\ \left| (-Δ)^{\frac {αr}2} (T_{α, t^α} f)(x) \right|> \varepsilon t^{s -r α}\right\}$$ be the set of ``bad'' points. To quantify its size, we introduce a class of extended nonnegative \emph{admissible set functions} $ν$ on the Borel $σ$-algebra $\mathcal{B}(\mathbb{R}^n\times [0, 1])$ and define, for any admissible function $ν$, the \emph{critical index} $ \varepsilon_{α, r, s,ν}(f):=\inf\{\varepsilon\in(0,\infty):\ ν(D_{α, r}(s,f,\varepsilon))<\infty\}.$ Our result shows that, for a broad class of subspaces $V\subset Λ_s$, including intersections of $Λ_s$ with Sobolev, Besov, Triebel--Lizorkin, and Besov-type spaces, there exists an admissible function $ν$ depending on $V$ such that $\varepsilon_{α, r, s,ν}(f)\sim \mathrm{dist}(f, V)_{Λ_s}.$ |
| title | Fractional Heat Semigroup Characterization of Distances from Functions in Lipschitz Spaces to Their Subspaces |
| topic | Functional Analysis Analysis of PDEs Classical Analysis and ODEs Primary 46E35, Secondary 26A16, 35K08, 42C40, 42E35 |
| url | https://arxiv.org/abs/2508.21269 |