临界 Sobolev-Hodge 空间中的有界代表元
Bounded Representatives in Critical Sobolev-Hodge Spaces
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中文总结 AI 辅助
针对 $n\geq2$、$1\leq\ell\leq n-1$ 且 $1<p<\infty$ 的情形，证明临界 Sobolev-Hodge 空间中每个 $v$ 存在满足 $du=dv$ 的有界代表元 $u$，通过归约为 Riesz 位势估计与 Hodge 投影等完成推导。
中文摘要 AI 辅助
设 $n\geq 2$，$1\leq \ell\leq n-1$，且 $1<p<\infty$。我们证明，每个 $v\in \dot W^{n/p,p}(\mathbb{R}^n;\Lambda^\ell)$ 都存在一个代表元 $u\in \dot W^{n/p,p}\cap L^\infty$，满足 $du=dv$，且 $\max\{|u|_{\dot W^{n/p,p}},|u|_{L^\infty}\}\lesssim |v|_{\dot W^{n/p,p}}$。等价地，$d[\dot W^{n/p,p}\Lambda^\ell]=d[(\dot W^{n/p,p}\cap L^\infty)\Lambda^\ell]$ 且商范数等价。证明将选取问题归约为 Riesz 位势的端点图估计与精确 Hodge 投影，核心分析输入是针对算子值齐次核的有限输入 Maz'ya--$\Phi$ 不等式。对 Riesz/Hodge 对，由投影核构造的非线性球剖面具有精确原子抵消性，且通过恒等式 $\int_{S^{n-1}}P(\theta)\\,d\bar{\sigma}=(\ell/n)\operatorname{Id}$ 具有强制性；频率局域化图闭性与 Hahn--Banach 对偶性最终给出有界代表元。
英文摘要
Let $n\geq 2$, $1\leq \ell\leq n-1$, and $1<p<\infty$. We prove that every $v\in \dot W^{n/p,p}(\mathbb{R}^n;Λ^\ell)$ has a representative $u\in \dot W^{n/p,p}\cap L^\infty$ with $du=dv$ and $\max{|u|{\dot W^{n/p,p}},|u|{L^\infty}}\lesssim |v|{\dot W^{n/p,p}}$. Equivalently, $d[\dot W^{n/p,p}Λ^\ell]=d[(\dot W^{n/p,p}\cap L^\infty)Λ^\ell]$ with equivalent quotient norms. The proof reduces the selection problem to an endpoint graph estimate for a Riesz potential and the exact Hodge projection. Its main analytic input is a finite-dimensional-input Maz'ya--$Φ$ inequality for operator-valued homogeneous kernels. For the Riesz/Hodge pair, a nonlinear spherical profile built from the projected kernel has exact atomic cancellation and is coercive by the identity $\int{S^{n-1}}P(θ),d\barσ=(\ell/n)\operatorname{Id}$. Frequency-localized graph closure and Hahn--Banach duality then return a bounded representative.
