Multilinear approximate identities generated by hypermetrics on spaces of homogeneous type

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.

💡 Research Summary

The paper introduces a novel framework for constructing multilinear integral operators by generalizing the classical metric‑based Newtonian potentials to (k + 1)‑ary hypermetrics. Starting from a quasi‑metric space (X, d) equipped with a doubling measure µ, the authors define a product quasi‑metric d_{k+1} on X^{k+1} by taking the supremum of the componentwise distances. The hypermetric ρ(x₀,…,x_k) is then defined as the distance from a point in X^{k+1} to the diagonal Δ_{k+1}. This geometric object measures how far a (k + 1)‑tuple is from being “coincident” and serves as the kernel’s argument.

A central object of study is the family of kernels \