Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 43, 2018, 121-145

GREEN KERNELS ASSOCIATED WITH RIESZ KERNELS

Bent Fuglede and Natalia Zorii

University of Copenhagen, Department of Mathematical Sciences
Universitetsparken 5, 2100 Copenhagen, Denmark; fuglede 'at' math.ku.dk

National Academy of Sciences of Ukraine, Institute of Mathematics
Tereshchenkivska 3, 01601, Kyiv-4, Ukraine; natalia.zorii 'at' gmail.com

Abstract. We study properties of the α-Green kernel g_D^α of order 0 < α ≤ 2 for a domain D ⊂ Rⁿ, n ≥ 3. This kernel is associated with the Riesz kernel |x – y|^α-n, x,y ∈ Rⁿ, in a manner particularly well known in the case α = 2. Besides the usual principles of potential theory, we establish for the α-Green kernel the property of consistency. This allows us to prove the completeness of the cone of positive measures μ on D with finite energy g_D^α(μ,μ) := ∫ g_D^α(x,y)dμ(x)dμ(y) in the topology defined by the energy norm ||μ||_gDα = √g_D^α(μ,μ), as well as the existence of the α-Green equilibrium measure for a relatively closed set in D of finite α-Green capacity. The main tool is a generalization of Cartan's theory of balayage (sweeping) for the Newtonian kernel to the α-Riesz kernels with 0 < α < 2.

2010 Mathematics Subject Classification: Primary 31C15.

Key words: α-Riesz balayage, α-Green kernels, consistency, α-Green equilibrium measure.

Reference to this article: B. Fuglede and N. Zorii: Green kernels associated with Riesz kernels. Ann. Acad. Sci. Fenn. Math. 43 (2018), 121-145.

Full document as PDF file

https://doi.org/10.5186/aasfm.2018.4305