An alternative concept of Riesz energy of measures with application to generalized condensers

Abstract

In view of a recent example of a positive Radon measure $\mu$ on a domain $D\subset\mathbb R^n$, $n\geqslant3$, such that $\mu$ is of finite energy $E_g(\mu)$ relative to the $\alpha$-Green kernel $g$ on $D$, though the energy of $\mu-\mu^{D^c}$ relative to the $\alpha$-Riesz kernel $|x-y|^{\alpha-n}$, $0<\alpha\leqslant2$, is not well defined (here $\mu^{D^c}$ is the $\alpha$-Riesz swept measure of $\mu$ onto $D^c=\mathbb R^n\setminus D$), we propose a weaker concept of $\alpha$-Riesz energy for which this defect has been removed. This concept is applied to the study of a minimum weak $\alpha$-Riesz energy problem over (signed) Radon measures on $\mathbb R^n$ associated with a (generalized) condenser ${\mathbf A}=(A_1,D^c)$, where $A_1$ is a relatively closed subset of $D$. A solution to this problem exists if and only if the $g$-capacity of $A_1$ is finite, which in turn holds if and only if there exists a so-called measure of the condenser $\mathbf A$, whose existence was analyzed earlier in different settings by Beurling, Deny, Kishi, Bliedtner, and Berg. Our analysis is based particularly on our recent result on the completeness of the cone of all positive Radon measures $\mu$ on $D$ with finite $E_g(\mu)$ in the metric determined by the norm $\|\mu\|_g:=\sqrt{E_g(\mu)}$. We also show that the pre-Hilbert space of Radon measures on $\mathbb R^n$ with finite weak $\alpha$-Riesz energy is isometrically imbedded into its completion, the Hilbert space of real-valued tempered distributions with finite energy, defined with the aid of Fourier transformation. This gives an answer in the negative to a question raised by Deny in 1950.