Necessary and sufficient conditions of the trilinear Stein-Weiss inequality

In this paper, we consider a trilinear weighted Hardy-Littlewood-Sobolev inequality as follows:

$$\left|{\displaystyle {\int }_{{\mathbb{R}}^n}{\displaystyle {\int }_{{\mathbb{R}}^n}{\displaystyle {\int }_{{\mathbb{R}}^n}\frac{{\prod }_{i=1}^3{f}_i(x_i)|x_i{|}^{-{\alpha }_{ii}}}{{\prod }_{1⩽j<k⩽3}|x_j-x_k{|}^{{\alpha }_{jk}}}d{x}_{1}d{x}_{2}d{x}_3}}}\right|⩽C\Vert f_1{\Vert }_{p_1}\Vert f_2{\Vert }_{p_2}\Vert f_3{\Vert }_{p_3},$$

which can be regarded as the natural trilinear form of the Stein-Weiss inequality. For $1⩽p_1,p_2,p_3⩽\mathrm{\infty }$, we systematically establish the trilinear Stein-Weiss inequality. In particular, when $1<p_1,p_2,p_3<\mathrm{\infty }$, by means of appropriate space decomposition, we give the necessary and sufficient conditions to characterize the trilinear Stein-Weiss inequality.