An Alternative Concept of Riesz Energy of Measures with Application to Generalized Condensers

TY - JOUR

T1 - An Alternative Concept of Riesz Energy of Measures with Application to Generalized Condensers

AU - Fuglede, Bent

AU - Zorii, Natalia

PY - 2019/8/15

Y1 - 2019/8/15

N2 - In view of a recent example of a positive Radon measure μ on a domain D⊂ ℝn, n⩾ 3 , such that μ is of finite energy Eg(μ) relative to the α-Green kernel g on D, though the energy of μ−μDc relative to the α-Riesz kernel |x − y|α−n, 0 < α⩽ 2 , is not well defined (here μDc is the α-Riesz swept measure of μ onto Dc= ℝn∖ D), 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 ℝn 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 ∥μ∥g:=Eg(μ). We also show that the pre-Hilbert space of Radon measures on ℝn 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.

AB - In view of a recent example of a positive Radon measure μ on a domain D⊂ ℝn, n⩾ 3 , such that μ is of finite energy Eg(μ) relative to the α-Green kernel g on D, though the energy of μ−μDc relative to the α-Riesz kernel |x − y|α−n, 0 < α⩽ 2 , is not well defined (here μDc is the α-Riesz swept measure of μ onto Dc= ℝn∖ D), 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 ℝn 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 ∥μ∥g:=Eg(μ). We also show that the pre-Hilbert space of Radon measures on ℝn 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.

KW - Condenser measure

KW - Condensers with touching plates

KW - Deny–Schwartz energy

KW - Minimum energy problems

KW - Standard and weak Riesz energy

KW - Tempered distributions

UR - http://www.scopus.com/inward/record.url?scp=85047220244&partnerID=8YFLogxK

U2 - 10.1007/s11118-018-9709-3

DO - 10.1007/s11118-018-9709-3

M3 - Journal article

AN - SCOPUS:85047220244

SN - 0926-2601

VL - 51

SP - 197

EP - 217

JO - Potential Analysis

JF - Potential Analysis

IS - 2

ER -