Optimal decay rates in Sobolev norms for singular values of integral operators

TL;DR

This paper introduces a new method to determine optimal decay rates of singular values for integral operators with Sobolev regularity, using Weyl's asymptotic formula and Neumann eigenfunctions, improving upon previous Fourier-based estimates.

Contribution

The paper develops a novel approach combining Weyl's asymptotic formula with domain-specific Neumann eigenfunctions to derive sharp decay estimates for singular values of integral operators.

Findings

Decay rates depend on dimension and Sobolev regularity

Optimal estimates are achieved using Neumann eigenfunctions

Results extend to real analytic kernels

Abstract

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 $(a,b) \times (a,b)$, where $(a,b)$ 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 $l^p$ norms, $p \geq 1$ or in case of faster decay due to regularity, $l^p$ quasi-norms, $0 < p < 1$. 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 $n^\alpha$ where for the $n$ -th singular value where $\alpha $ 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 $n^\alpha$, we use the singular values of the kernel obtained by differentiation. While in \cite{birman1977estimates, delgado2021schatten} $l^p$ 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.