Potential theory with multivariate kernels

Dmitriy Bilyk*, Damir Ferizović, Alexey Glazyrin, Ryan William Matzke, Josiah Park, Oleksandr Vlasiuk

*Corresponding author for this work

Research output: Contribution to journalArticle


In the present paper we develop the theory of minimization for energies with multivariate kernels, i.e. energies, in which pairwise interactions are replaced by interactions between triples or, more generally, n-tuples of particles. Such objects, which arise naturally in various fields, present subtle differences and complications when compared to the classical two-input case. We introduce appropriate analogues of conditionally positive definite kernels, establish a series of relevant results in potential theory, explore rotationally invariant energies on the sphere, and present a variety of interesting examples, in particular, some optimization problems in probabilistic geometry which are related to multivariate versions of the Riesz energies.
Original languageEnglish
Pages (from-to)2907-2935
Number of pages23
JournalMathematische Zeitschrift
Publication statusPublished - 2022


  • Classical Analysis
  • Functional Analysis

ASJC Scopus subject areas

  • Analysis


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