$\mathbb{R}^n$中无可微性要求函数在一般集上的迹
Traces on General Sets in $\mathbb{R}^n$ for Functions with no Differentiability Requirements
摘要：
本文研究可积但在其定义域内不需要具有任何可微性的函数的迹理论。此外，域可以具有不规则的边界，具有尖点状特征，并且余维不一定等于1，甚至是整数。给定$Ω\subseteq\mathbb{R}^n$和$Γ\subseteq\partialΩ$，我们引入了一个函数空间$\mathscr{N}^{s(\cdot),p}(Ω)\subseteq L^p_{\text{loc}}(Ω)$，可以确定一个定义良好的跟踪操作。当接近$Γ$时，$\mathscr{N}^{s(\cdot),p}(Ω)$中的成员资格约束函数值的振荡，但并不意味着远离$Γ$的任何规律性。在$Ω$和$Γ$连通的假设下，我们构造了一个从$\mathscr{N}^{s(\cdot),p}(Ω)$到$Γ$上可测函数空间的线性迹算子.例如，所有$1$边的非切向可达域都满足连通性假设.如果$Γ$是上Ahlfors正则的，则迹是进入Sobolev-Slobodeckij空间的连续算子。如果进一步假设$Γ=\partialΩ$和是下Ahlfors正则的，则迹具有标准Lebesgue点性质。为了证明结果的一般性，我们构造了具有满足主域假设的$t>1$维Ahlfors正则$Γ\subseteq\partialΩ$的$Ω\subseteq\mathbb{R}^2$，但是$Γ$无处可求，并且对于$Γ$中每个点的每个邻域，在该邻域内存在只能切向访问的边界点。
英文摘要：
This paper is concerned with developing a theory of traces for functions that are integrable but need not possess any differentiability within their domain. Moreover, the domain can have an irregular boundary with cusp-like features and codimension not necessarily equal to one, or even an integer. Given $Ω\subseteq\mathbb{R}^n$ and $Γ\subseteq\partialΩ$, we introduce a function space $\mathscr{N}^{s(\cdot),p}(Ω)\subseteq L^p_{\text{loc}}(Ω)$ for which a well-defined trace operator can be identified. Membership in $\mathscr{N}^{s(\cdot),p}(Ω)$ constrains the oscillations in the function values as $Γ$ is approached, but does not imply any regularity away from $Γ$. Under connectivity assumptions between $Ω$ and $Γ$, we produce a linear trace operator from $\mathscr{N}^{s(\cdot),p}(Ω)$ to the space of measurable functions on $Γ$. The connectivity assumptions are satisfied, for example, by all $1$-sided nontangentially accessible domains. If $Γ$ is upper Ahlfors-regular, then the trace is a continuous operator into a Sobolev-Slobodeckij space. If $Γ=\partialΩ$ and is further assumed to be lower Ahlfors-regular, then the trace exhibits the standard Lebesgue point property. To demonstrate the generality of the results, we construct $Ω\subseteq\mathbb{R}^2$ with a $t>1$-dimensional Ahlfors-regular $Γ\subseteq\partialΩ$ satisfying the main domain hypotheses, yet $Γ$ is nowhere rectifiable and for every neighborhood of every point in $Γ$, there exists a boundary point within that neighborhood that is only tangentially accessible.
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