On the Positive-Definiteness of an Anisotropic Operator

We study the positive-definiteness of a family of $L^2(\mathbf{R})$ integral operators with kernel $K_{t, a}(x, y) = (1 + (x - y)^2 + a(x^2 + y^2)^t)^{-1}$, with $t > 0$ and $a > 0$. When $0 < t \le 1$, the known theory of positive-definite kernels ensures that the operator is positive-definite; when $t > 1$, constructions disprove positive-definiteness for many $(t, a)$-pairs.