We construct whole-space extensions of functions in a fractional Sobolev space of order \(s\in (0,1)\) and integrability \(p\in (0,\infty )\) on an open set O which vanish in a suitable sense on a portion D of the boundary \({{\,\mathrm{\partial \!}\,}}O\) of O. The set O is supposed to satisfy the so-called interior thickness condition in \({{\,\mathrm{\partial \!}\,}}O {\setminus } D\), which is much weaker than the global interior thickness condition. The proof works by means of a reduction to the case \(D=\emptyset \) using a geometric construction.

中文翻译：

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局部消失迹条件下分数阶Sobolev空间的扩展问题

我们在一个消失的开放集O上以分数\（s \ in（0,1）\）和可积性\（p \ in（0，\ infty）\）的分数Sobolev空间构造函数的全空间扩展。上的部分的合适的感d边界\（{{\，\ mathrm {\局部\！} \，}}ø\）的Ó。集合O应该满足\（{{\，mathrm {\ partial \！} \，}} O {\ setminus} D \）中的所谓内部厚度条件，该条件比全局内部弱得多厚度条件。该证明通过使用几何构造将情况简化为\（D = \ emptyset \）来工作。