Boundedness properties of the bilinear fractional integral operators induced by hypermetrics of third order
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915947851284480 |
|---|---|
| author | Aimar, Hugo Gómez, Ivana Toledo, Joaquín |
| author_facet | Aimar, Hugo Gómez, Ivana Toledo, Joaquín |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_19739 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Boundedness properties of the bilinear fractional integral operators induced by hypermetrics of third order Aimar, Hugo Gómez, Ivana Toledo, Joaquín Classical Analysis and ODEs 42B20, 47H60, 47G10, 42B25 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. |
| title | Boundedness properties of the bilinear fractional integral operators induced by hypermetrics of third order |
| topic | Classical Analysis and ODEs 42B20, 47H60, 47G10, 42B25 |
| url | https://arxiv.org/abs/2604.19739 |