Necessary and sufficient conditions of the trilinear Stein-Weiss inequality

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Abstract

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

$\left|\int_{\mathbb{R}^{n}} \int_{\mathbb{R}^{n}} \int_{\mathbb{R}^{n}} \frac{\prod_{i=1}^{3} f_{i}\left(x_{i}\right)\left|x_{i}\right|^{-\alpha_{i i}}}{\prod_{1 \leqslant j<k \leqslant 3}\left|x_{j}-x_{k}\right|^{\alpha_{j k}}} d x_{1} d x_{2} d x_{3}\right| \leqslant C\left\|f_{1}\right\|_{p_{1}}\left\|f_{2}\right\|_{p_{2}}\left\|f_{3}\right\|_{p_{3}},$

which can be regarded as the natural trilinear form of the Stein-Weiss inequality. For 1 ⩽ p₁, p₂, p₃ ⩽ ∞, we systematically establish the trilinear Stein-Weiss inequality. In particular, when 1 < p₁, p₂, p₃ < ∞, by means of appropriate space decomposition, we give the necessary and sufficient conditions to characterize the trilinear Stein-Weiss inequality.

Keywords

Stein-Weiss inequalities / boundedness of trilinear functionals / necessary and sufficient conditions

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