LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-09-23 (gpt-6-sol)

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Radial mollification fixes local mean-value functions

Statement

Assume the Axiom of Countable Choice. Let $u \in C (Ω)$ have the spherical mean-value property, and let $(ρ_{ε})$ be a radial mollifier family as in A radial mollifier family in Rn. Then $(u * ρ_{ε}) (x) = u (x)$ whenever $B_{ε} (x) ⋐ Ω$ .

Proof

Given: Countable Choice, $u$ has the local spherical mean-value property, and $B_{ε} (x) ⋐ Ω$ .

1.1

Under Countable Choice, Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma gives $(u * ρ_{ε}) (x) = ω_{n - 1} \int_{0}^{ε} q_{ε} (t) t^{n - 1} M_{u} (x, t) d t$ [given].

2.1

Substitute $M_{u} (x, t) = u (x)$ and use $\int ρ_{ε} = 1$ to obtain $(u * ρ_{ε}) (x) = u (x)$ [given]. ∎

Depends on

Dependency tree · 2 levels

Sources

PDE source treatment