Subharmonic Kernels and Energy Minimizing Measures, with Applications to the Flat Torus

Abstract

We study the minimization of the energy integral $I_K(\mu) = \int_{\Omega} \int_{\Omega} K(x,y) d\mu(x) d\mu(y)$ over all Borel probability measures $\mu$, where $(\Omega,\rho)$ is a compact connected metric space and $K:\Omega^2 \to [0,\infty]$ is continuous in the extended sense. We focus on kernels $K$ which are subharmonic, which we define so that the potential $U_K^\mu(x) = \int_{\Omega} K(x,y) d\mu(y)$ satisfies a maximum principle on $\Omega\setminus{\rm supp}{\mu}$. This extends the classical electrostatics minimization problem for logarithmic energy $\int_{\Omega}\int_{\Omega}\log\left(\frac{1}{||x-y||}\right)$, which is used heavily as a tool in approximation theory. Using properties of minimizing measures, we show that if the singularities of the subharmonic kernel $K$ are such that $K$ is regular, then $K$ is positive definite, and $\mu$ is a minimizing measure if and only if its potential is constant (outside of a small exceptional set).We then apply this result to group invariant kernels on compact homogeneous manifolds. In this case, the uniform measure $\sigma$ has constant potential, so subharmonicity implies that this is the minimizing measure. Finally, we look at the case of the $d$-dimensional flat torus $T^d$. We use our results to see that the Riesz kernel $K_s(x,y) = {\rm sign}(s)\rho(x,y)^{-s}$ is minimized by $\sigma$ (and thus positive definite) when $d > s \geq d-2$. Additionally, the positive definiteness gives us a condition which implies that the multivariate Fourier series of a function $f:[0,\pi]^d \to [0,\infty]$ has nonnegative coefficients.