We prove an optimal extension and trace theorem for Sobolev spaces of nonlocal operators. The extension is given by a suitable Poisson integral and solves the corresponding nonlocal Dirichlet problem. We give a Douglas-type formula for the quadratic form of the Poisson extension.

中文翻译：

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非本地运营商的扩展和跟踪

我们证明了非局部算子的Sobolev空间的最优扩展和跟踪定理。该扩展由合适的泊松积分给出，并解决了相应的非局部Dirichlet问题。我们给出泊松扩展的二次形式的道格拉斯型公式。