2025/08/29 by Chang‐Feng Dai, Dai, Feng, Eero Saksman +7
Mathematics · #35K08 #42C40 #42E35 #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #Approximation Theory and Sequence Spaces #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Differential Equations Analysis #Primary 46E35 #Secondary 26A16
paper · pdf · doi:10.48550/arxiv.2508.21269
openalex publication_date 2025/08/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Λs denote the inhomogeneous Lipschitz space of order s∈(0,∞) on ℝn. This article characterizes the distance d(f, V)Λs: = infg∈ V ‖f-g‖Λs from a function f∈ Λs to a non-dense subspace V⊂ Λs via the fractional semigroup \Tα, t: =e^-t (-Δ)α/2: t∈ (0, ∞)\ for any α∈(0,∞). Given an integer r >s/α, a uniformly bounded continuous function f on ℝn belongs to the space Λs if and only if there exists a constant λ∈(0,∞) such that |(-Δ)^\frac αr2 (Tα, tα f)(x) |≤ λts -rα for any x∈ℝn and t∈ (0, 1]. The least such constant is denoted by λ α, r, s(f). For each f∈ Λs and 0<ε< λα,r, s(f), let Dα, r(s,f,ε):=\ (x,t)∈ ℝn× (0,1]: | (-Δ)^\frac αr2 (Tα, tα f)(x) |gt; ε ts -r α\ be the set of ``bad'' points. To quantify its size, we introduce a class of extended nonnegative admissible set functions ν on the Borel σ-algebra B(ℝn× [0, 1]) and define, for any admissible function ν, the critical index εα, r, s,ν(f):=inf\ε∈(0,∞): ν(Dα, r(s,f,ε))<∞\. Our result shows that, for a broad class of subspaces V⊂ Λs, including intersections of Λs with Sobolev, Besov, Triebel--Lizorkin, and Besov-type spaces, there exists an admissible function ν depending on V such that εα, r, s,ν(f)∼ dist(f, V)Λs.