We prove a two-parameter family of second order geometric Rellich type equalities on the half-space \( \mathbb {R} _{+}^{n}\). We then use it to derive several refined geometric Rellich type inequalities. We also prove an improved version of the Rellich–Sobolev–Maz’ya type inequality.

中文翻译：

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关于半空间上的二阶几何Rellich不等式的一个注记

我们证明了半空间\（\ mathbb {R} _ {+} ^ {n} \）上的二参数几何Rellich型等式的两参数族。然后，我们使用它来推导几个精炼的几何Rellich型不等式。我们还证明了Rellich-Sobolev-Maz'ya型不等式的改进版本。