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 ${△}_3=\{(x,x,x)\in X^3:x\in X\}$ 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 $\Vert T^{\gamma }(f,g){\Vert }_{p_3}\le C\Vert f{\Vert }_{p_1}\Vert g{\Vert }_{p_2}$ 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.

9 pages