On eigenvalues of differentiable positive definite kernels (Q1280553)

The authors consider compact integral operators in \(L^2(0,1)\) with a Hermitian kernel \(K(x,y)\), which is positive definite and has continuous partial derivatives with respect to \(y\) up to order \(p\). It is proven that the eigenvalues \(\lambda_n\), have the asymptotics, \(\lambda_n=o(n^{-p-1}), n\to \infty\). If, in addition, the \(p-\)th derivative is Lipschitz continuous in \(y\) of order \(0<\alpha\leq 1\), then \(\lambda_n =O(n^{-p-1-\alpha}), n\to \infty\).