The regularity of integration kernels forces decay rates of singular values of associated integral operators. This is well-known for symmetric operators with kernels defined on , where is an interval. Over time, many authors have studied this case in detail . The case of spheres has also been resolved . A few authors have examined the higher dimensional case or the case of manifolds . Typically, these authors have provided decay estimates of singular values in norms, or in case of faster decay due to regularity, quasi-norms, . With that approach, it is straightforward to show that their estimates are optimal using periodic kernels obtained from Fourier series. Our new approach for deriving decay estimates of these singular values uses Weyl's asymptotic formula for Neumann eigenvalues that we combine to an appropriately defined inverse Laplacian. We obtain decay estimates in the form where for the -th singular value where depends on dimension and on the Sobolev regularity of the kernel. Since we are interested in optimal estimates in case of regular kernels, instead of writing an upper bound by a constant times , we use the singular values of the kernel obtained by differentiation. While in estimates were proven to be optimal by simply considering periodic Fourier series, these Fourier series do not provide sharp results for our estimates. Instead, series of Neumann eigenfunctions for the Laplacian that are specific to the domain of interest are used to prove that our decay estimates are optimal. Finally, we cover the case of real analytic kernels where we are also able to derive optimal estimates.

中文翻译：

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积分算子奇异值的 Sobolev 范数中的最优衰减率

积分核的规律性迫使相关积分算子的奇异值的衰减率。这对于具有定义在 上的内核的对称运算符来说是众所周知的，其中 是一个区间。随着时间的推移，许多作者对这个案例进行了详细的研究。球体的情况也已得到解决。一些作者研究了高维情况或流形的情况。通常，这些作者提供了范数中奇异值的衰减估计，或者在由于规律性、准范数而导致更快衰减的情况下。通过这种方法，可以直接证明使用从傅里叶级数获得的周期核，他们的估计是最佳的。我们推导这些奇异值的衰减估计的新方法使用 Weyl 的诺依曼特征值渐近公式，我们将其与适当定义的逆拉普拉斯算子结合起来。我们以以下形式获得衰减估计： 其中，对于第 - 个奇异值，其中取决于维度和核的 Sobolev 正则性。由于我们对常规核的最优估计感兴趣，因此我们使用通过微分获得的核的奇异值，而不是通过常数 times 编写上限。虽然通过简单地考虑周期性傅里叶级数来估计被证明是最佳的，但这些傅里叶级数并没有为我们的估计提供清晰的结果。相反，使用特定于感兴趣域的拉普拉斯算子的一​​系列诺伊曼本征函数来证明我们的衰减估计是最优的。最后，我们介绍了真实分析核的情况，我们也能够得出最佳估计。