Fractional integrals associated with Zygmund dilations

We study a family of fractional integral operators defined in ${\mathbb{R}}^3$ whose kernels are distributions associated with Zygmund dilations: $(x_1,x_2,x_3)\to ({\delta }_1{x}_1,{\delta }_2{x}_2,{\delta }_1{\delta }_2{x}_3)$ for ${\delta }_1,{\delta }_2>0$ having singularity on every coordinate subspace. As a result, we obtain a Hardy-Littlewood-Sobolev type inequality.