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

Abstract

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 $\rho(x,y,z)$, 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 $(X,d,\mu)$ is $\eta$-Ahlfors regular for some $\eta$ positive, the function $\rho(x,y,z)$ generates kernels for bilinear operators of the type $T^{\gamma}(f,g)(x)=\iint_{X\times X}\rho(x,y,z)^{-\gamma}f(y)g(z)d\mu(y)d\mu(z)$, for a given positive $\gamma$. In the setting of $\eta$-Ahlfors regular space, the power $-\gamma=-2\eta$ of $\rho(x,\cdot,\cdot)$ provides the natural singularity for this family of kernels. In this paper we consider the fractional integral rank $0<\gamma<2\eta$. We prove boundedness properties of the type $\|T^{\gamma}(f,g)\|_{p_3}\leq C\|f\|_{p_1}\|g\|_{p_2}$ for adequate values of the exponents $p_1,p_2$ and $p_3$. The proof is based on three upper bounds for $T^{\gamma}(f,g)$ in terms of the classical linear fractional Riesz operators $I_{\eta-\frac{\gamma}{2}}$, using the linear Hardy-Littlewood-Sobolev inequality.