On eigenvalues of differentiable positive definite kernels

If a positive definite kernelK(x, y) has thepth order partial derivative (∂p/∂yp)K(x,y) continuous on the square [0,1]2, we show that the eigenvalues of the integral operator generated byK(x, y) are asymptoticallyo(1/np+1). We also obtain the anticipated asymptotic estimate when (∂p/∂yp)K(x,y) satisfies further a Lipschitz condition iny of order 0<α≤1. These results, which extend some classical estimates of I. Fredholm and H. Weyl under the additional positive definiteness assumption, are based on two interesting inequalities of K. Fan.