标题： 迹算子的核通过精细连续性 显示英文标题

标题： Kernels of trace operators via fine continuity

摘要： 我们研究分数阶Sobolev空间$H_p^\alpha(\mathbb{R}^n)$元素在$\mathbb{R}^n$的闭子集$\Gamma$上的迹，这些迹是适当测度$\mu$的支持。 我们证明，如果这些测度满足局部上密度条件，则拟连续表示在$\Gamma$上准处处消失当且仅当它们在$\Gamma$上几乎处处消失。 我们利用这个结果刻画了从$H_p^\alpha(\mathbb{R}^n)$到$\mu$-函数等价类空间的迹算子映射的核，其性质为$C_c^\infty(\mathbb{R}^n\setminus \Gamma)$在$H_p^\alpha(\mathbb{R}^n)$中的闭包。 测度不必满足加倍条件。 具体而言，集合 $\Gamma$ 可以是具有不同豪斯多夫维数的闭集的有限并集。 我们提供分数 Sobolev 空间$H_p^\alpha(\Omega)$在满足测度密度条件的区域$\Omega\subset \mathbb{R}^n$上的相应结果。 显示英文摘要

摘要： We study traces of elements of fractional Sobolev spaces $H_p^\alpha(\mathbb{R}^n)$ on closed subsets $\Gamma$ of $\mathbb{R}^n$, given as the supports of suitable measures $\mu$. We prove that if these measures satisfy localized upper density conditions, then quasi continuous representatives vanish quasi everywhere on $\Gamma$ if and only if they vanish $\mu$-almost everywhere on $\Gamma$. We use this result to characterize the kernel of the trace operator mapping from $H_p^\alpha(\mathbb{R}^n)$ into the space of $\mu$-equivalence classes of functions on $\Gamma$ as the closure of $C_c^\infty(\mathbb{R}^n\setminus \Gamma)$ in $H_p^\alpha(\mathbb{R}^n)$. The measures do not have to satisfy a doubling condition. In particular, the set $\Gamma$ may be a finite union of closed sets having different Hausdorff dimensions. We provide corresponding results for fractional Sobolev spaces $H_p^\alpha(\Omega)$ on domains $\Omega\subset \mathbb{R}^n$ satisfying the measure density condition.