An Alternative Concept of Riesz Energy of Measures with Application to Generalized Condensers
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Bent Fuglede, Natalia Zorii
In view of a recent example of a positive Radon measure μ on a domain (Formula presented.), (Formula presented.), such that μ is of finite energy Eg(μ) relative to the α-Green kernel g on D, though the energy of (Formula presented.) relative to the α-Riesz kernel |x − y|α−n, (Formula presented.), is not well defined (here (Formula presented.) is the α-Riesz swept measure of μ onto (Formula presented.)), we propose a weaker concept of α-Riesz energy for which this defect has been removed. This concept is applied to the study of a minimum weak α-Riesz energy problem over (signed) Radon measures on (Formula presented.) associated with a (generalized) condenser A = (A1,Dc), where A1 is a relatively closed subset of D. A solution to this problem exists if and only if the g-capacity of A1 is finite, which in turn holds if and only if there exists a so-called measure of the condenser 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 μ on D with finite Eg(μ) in the metric determined by the norm (Formula presented.). We also show that the pre-Hilbert space of Radon measures on (Formula presented.) with finite weak α-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.
|Status||E-pub ahead of print - 2019|