Joaquín Toledo | Searcharxiv

arXiv2026

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.

math.CA↗

arXiv2026

Boundedness properties of the bilinear fractional integral operators induced by hypermetrics of third order

We introduce a natural bilinear fractional integral type operator induced by a third order hypermetric on Ahlfors regular quasi-metric spaces. Given a quasi-metric space $(X,d)$ the function $ρ(x,y,z)$, defined as the distance, in $X^3$, of $(x,y,z)$ to the diagonal $\bigtriangleup_3=\{(x,x,x)\in X^3:x\in X\}$ is said to be a third order hypermetric in $X$. When $(X,d)$ is a Euclidean space or, more generally, when $(X,d,μ)$ is $η$-Ahlfors regular for some $η$ positive, the function $ρ(x,y,z)$ generates kernels for bilinear operators of the type $T^γ(f,g)(x)=\iint_{X\times X}ρ(x,y,z)^{-γ}f(y)g(z)dμ(y)dμ(z)$, for a given positive $γ$. In the setting of $η$-Ahlfors regular space, the power $-γ=-2η$ of $ρ(x,\cdot,\cdot)$ provides the natural singularity for this family of kernels. In this paper we consider the fractional integral rank $0<γ<2η$. We prove boundedness properties of the type $\|T^γ(f,g)\|_{p_3}\leq C\|f\|_{p_1}\|g\|_{p_2}$ for adequate values of the exponents $p_1,p_2$ and $p_3$. The proof is based on three upper bounds for $T^γ(f,g)$ in terms of the classical linear fractional Riesz operators $I_{η-\fracγ{2}}$, using the linear Hardy-Littlewood-Sobolev inequality.

math.CA↗