AI 中文总结
该研究针对$\boldsymbol{\text{R}}^n$上带非负权重的三线性Stein-Weiss不等式，在特定假设下证明极值可达性，结合对称递减重排等方法推导相关方程并分析其不变性。
AI 中文摘要
我们研究$\boldsymbol{\text{R}}^n$上三线性Stein-Weiss不等式的极值化子。在已知的有界区域内，我们在两个附加假设下证明其可达性：所有六个权重指数均为非负，且至少有一对Lebesgue指数是可容许的。该证明结合对称递减重排与对数径向约化，得到$\boldsymbol{\text{R}}$上的平移不变双线性算子，其核属于$L^1(\boldsymbol{\text{R}}^2)$。采用共同尺度紧性论证排除两个自变量的相对分离，从而得到范数可达性。随后我们推导欧拉-拉格朗日方程组，在完全对称情形下，每个归一化非负极值三元组都是对角的。最后，我们确定所得标量方程在尺度指数处的原点中心Kelvin不变性，并记录无权重共形例子。
英文摘要
We study extremizers for a trilinear Stein-Weiss inequality on $\mathbb{R}^n$. Within the known boundedness region, we prove attainment under two additional assumptions: all six weight exponents are nonnegative, and at least one pair of Lebesgue exponents is admissible. The proof combines symmetric decreasing rearrangement with a logarithmic radial reduction to a translation-invariant bilinear operator on $\mathbb{R}$ whose kernel belongs to $L^1\left(\mathbb{R}^2\right)$. A common-scale compactness argument rules out relative separation of the two arguments and yields norm attainment. We then derive the Euler-Lagrange system. In the fully symmetric case, every normalized nonnegative extremizing triple is diagonal. Finally, we establish the origin-centered Kelvin invariance of the resulting scalar equation at the scaling exponent and record the unweighted conformal example.
