We consider Sobolev problems (problems for an elliptic operator on a closed manifold with conditions on a closed submanifold) for the case in which these conditions are of nonlocal nature and include weighted spherical means of the unknown function over spheres of a given radius. For such problems, we establish a criterion for the Fredholm property and, in some special cases, obtain index formulas.

中文翻译：

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具有球形平均条件的Sobolev问题和量化的规范变换轨迹

对于这些条件具有非局部性质，并且包括给定半径的球面上的未知函数的加权球面均值，我们考虑了Sobolev问题（椭圆形算子在闭合流形上的问题，这些条件在闭合子流形上）。对于此类问题，我们为Fredholm属性建立了标准，并在某些特殊情况下，获得了指数公式。