On determination of positive-definiteness for an anisotropic operator

Abstract

We study the positive-definiteness of a family of $L^2(\mathbb {R})$ integral operators with kernel $K_{t, a} (x, y) = \pi ^{-1} (1 + (x - y)^2+ a(x^2 + y^2)^t)^{-1}$, for $t > 0$ and $a > 0$. For $0 < t \le 1$ and $a > 0$, the known theory of positive-definite kernels and conditionally negative-definite kernels confirms positive-definiteness. For $t > 1$ and a sufficiently large, the integral operator is not positive-definite. For t not an integer, but with integer odd part, the integral operator is not positive-definite.