Estimates of singular values of integral operators with analytic kernels (Q1204948)

The paper is concerned with integral operators of the form \[ (Kf)(x)= \int^ 1_ 0 k(x, y) f(y) dy \] on the space \(L^ 2(0, 1)\), or similar operators when Lebesgue measure is substituted by a Borel measure on \(\mathbb{R}\). Ten theorems are stated in this paper (some of them with proof). For instance, Theorem 10 deals with kernels \(k(x, y)\) which can be analytically extended to a domain of the form \(\{z; |\arg z|< \lambda, | z|< \mathbb{R}\}\), \(0< \lambda\leq \pi\), \(\mathbb{R}> 1\). If this extension satisfies an estimate of the form \[ \int^ 1_ 0 | k(z, y)|^ 2dy\leq c| z|^{- 2p}\quad (0\leq p< 1/2) \] in the extended domain, then an asymptotic estimate is obtained for the singular numbers of \(K\).