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: $(x_1, x_2, x_3) \rightarrow (\delta_1 x_1, \delta_2 x_2, \delta_1\delta_2 x_3)$ for $\delta_1,\delta_2>0$ having singularity on every coordinate subspace. As a result, we obtain a Hardy-Littlewood-Sobolev type inequality.