On determination of positive-definiteness for an anisotropic operator

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.