Abstract The Branson–Gover operators are conformally invariant differential operators of even degree acting on differential forms. They can be interpolated by a holomorphic family of conformally invariant integral operators called fractional Branson–Gover operators. For Euclidean spaces we show that the fractional Branson–Gover operators can be obtained as Dirichlet-to-Neumann operators of certain conformally invariant boundary value problems, generalizing the work of Caffarelli–Silvestre for the fractional Laplacians to differential forms. The relevant boundary value problems are studied in detail and we find appropriate Sobolev type spaces in which there exist unique solutions and obtain the explicit integral kernels of the solution operators as well as some of their properties.

中文翻译：

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与分数式 Branson-Gover 算子相关的扩展问题

摘要 Branson-Gover 算子是作用于微分形式的偶次共形不变微分算子。它们可以通过称为分数布兰森-戈弗算子的保形不变积分算子的全纯族进行插值。对于欧几里德空间，我们表明分数 Branson-Gover 算子可以作为某些保形不变边值问题的 Dirichlet-to-Neumann 算子获得，将 Caffarelli-Silvestre 对分数拉普拉斯算子的工作推广到微分形式。详细研究了相关的边值问题，找到了存在唯一解的合适的Sobolev类型空间，并获得了解算子的显式积分核及其一些性质。