Multilinear approximate identities generated by hypermetrics on spaces of homogeneous type

TL;DR

This work develops a multilinear approximation framework for the multilinear identity on spaces of homogeneous type via hypermetrics. It introduces kernels built from the hypermetric $\rho$ and analyzes their maximal control through a Hardy–Littlewood type multilinear operator, establishing $L^{p_1}\times\cdots\times L^{p_k}\to L^1$ boundedness under $\sum 1/p_j=1$. The main result proves that as $\varepsilon\to0$, the multilinear approximate identities converge pointwise almost everywhere to the true multilinear identity:

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.

Theorems & Definitions (17)

• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Proposition 3.3
• proof