*Most theoretical papers concerning kernels assume that they are given a positive definite kernel. In this question, we want to show that a specific kernel is positive definite.*

We are interested in the following kernel defined for $0\leq x,y \leq 1$: $$K(x,y) = (x+y)^{3/2} - |x-y|^{3/2}.$$ Numerical simulations have shown that this kernel is positive definite, that is, for all non trivial $\phi : (0,1) \mapsto \mathbb{R}$, we have the following inequality: $$ \int_0^1 \int_0^1 K(x,y) \phi(x) \phi(y) > 0.$$ Can this fact be proved? Where should I look for references?