Multilinear approximate identities generated by hypermetrics on spaces of homogeneous type

TL;DR

This paper explores multilinear approximation of identities using potentials generated by hypermetrics in spaces of homogeneous type, extending classical metric-based harmonic analysis to multilinear and hypermetric contexts.

Contribution

It introduces a framework for multilinear approximation using hypermetrics, generalizing classical potential theory to multilinear operators in spaces of homogeneous type.

Findings

Development of multilinear kernels from hypermetrics

Extension of harmonic analysis to multilinear operators

Approximation of identities in generalized metric spaces

Abstract

The classical Newtonian potentials, defined in terms of metrics, give rise to the basic family of kernels defining linear integral operators and posing the fundamental problems of linear harmonic analysis. When the binary character of a metric on a set is naturally generalized to the $(k+1)$-ary character of hypermetric on the set, we obtain families of kernels of $k+1$ variables leading to multilinear integral operators of order $k$ or $k$-linear operators. In this paper we consider the problem of multilinear approximation to the multilinear identity through potentials built on hypermetrics in the general setting of spaces of homogeneous type.