C²函数边界值的Sharp有限性原理：长版本
Sharp Finiteness Principles for the boundary values of $C^2$-functions
AI总结：
针对Rⁿ中区域Ω的C²函数边界值，证明了有限性原理，即f为C²函数边界迹的条件可限定为含特定几何可见性性质的N=3·2ⁿ⁻¹点子集的验证，改进了原判据。
摘要：
设Ω为Rⁿ中具有边界∂Ω的区域，我们证明了关于C²(Ω)函数边界值的有限性原理：若存在常数λ>0，使得对∂Ω中任意至多包含N=3·2ⁿ⁻¹个点的集合E，都存在函数F_E∈C²(Ω)满足||F_E||_{C²(Ω)}≤λ，且其在∂Ω上的迹与f在E上一致，则函数f:∂Ω→R是某函数F∈C²(Ω)在边界上的迹。此外，我们改进了该原理，表明此判据可限定为∂Ω中相对于Ω具有特定几何“可见性”性质的N点子集E。
英文摘要：
Let $\Omega$ be an arbitrary domain in ${\bf R}^n$, and let $C^2(\Omega)$ be the space of all functions whose partial derivatives of all orders up to two are continuous and bounded on $\Omega$. We prove the following Finiteness Principle for the boundary values of $C^2(\Omega)$-functions: A function $f$ defined on the boundary of $\Omega$ is the trace of a function $F\in C^2(\Omega)$ provided there exists $\lambda>0$ such that, for every set $E$ of at most $N=3\cdot 2^{n-1}$ boundary points, there exists a function $F_E\in C^2(\Omega)$ with $\|F_E\|_{C^2(\Omega)}\le\lambda$ whose trace coincides with $f$ on $E$. Moreover, $F$ can be chosen with $\|F\|_{C^2(\Omega)}\le\gamma\lambda$ where $\gamma$ depends only on $n$. We also refine this principle by showing that it suffices to consider $N$-point sets $E$ with specific geometric ``visibility'' properties relative to $\Omega$.
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