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Optimal decay rates in Sobolev norms for singular values of integral operators | | Darko Volkov
; | | Date: |
1 Sep 2023 | | 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 $(a,b) imes (a,b)$, where $(a,b)$ is an interval.
Over time, many authors have studied this case in detail cite{GohbergKrein,
ha1986, chang1999eigenvalues, little1984eigenvalues, hille1931characteristic}.
The case of spheres has also been solved cite{castro2012eigenvalue}. However,
few authors have examined the higher dimensional case or the case of manifolds
cite{birman1977estimates}. Our new approach for deriving decay estimates of
these singular values uses Weyl’s asymptotic formula for Neumann eigenvalues
cite{weyl1912asymptotische} that we combine to an appropriately defined
inverse Laplacian. As we work with Sobolev norms, we are able to prove that our
estimates are optimal if the boundary of the underlying open set is smooth
enough. If this boundary is only Lipschitz regular, we still find quasi
optimality. Finally we cover the case of real analytic kernels where we are
also able to derive optimal estimates. | | Source: | arXiv, 2309.00153 | | Services: | Forum | Review | PDF | Favorites | |
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