The Hardy-Rellich inequality in the whole space with the best constant was firstly proved by Tertikas and Zographopoulos in Adv. Math. (2007) in higher dimensions N ⩾ 5. Then it was extended to lower dimensions N ∈ {3, 4} by Beckner in Forum Math. (2008) and Ghoussoub-Moradifam in Math. Ann. (2011) by applying totally different techniques.In this note, we refine the method implemented by Tertikas and Zographopoulos, based on spherical harmonics decomposition, to give an easy and compact proof of the optimal Hardy–Rellich inequality in any dimension N ⩾ 3. In addition, we provide minimizing sequences which were not explicitly mentioned in the quoted papers in lower dimensions N ∈ {3, 4}, emphasizing their symmetry breaking. We also show that the best constant is not attained in the proper functional space.

中文翻译：

---

Hardy-Rellich 不等式在任何维度上的新证明

具有最佳常数的整个空间的 Hardy-Rellich 不等式由 Tertikas 和 Zographopoulos 在 Adv. 中首先证明。数学。（2007）更高维度ñ⩾ 5. 然后扩展到更低的维度ñ∈ {3, 4} by Beckner 在数学论坛。（2008 年）和 Ghoussoub-Moradifam 在数学。安。（2011）通过应用完全不同的技术。在本说明中，我们改进了 Tertikas 和 Zographopoulos 实施的方法，基于球谐分解，为任何维度上的最优 Hardy-Rellich 不等式提供了一个简单而紧凑的证明ñ⩾ 3. 此外，我们提供了在低维引用论文中未明确提及的最小化序列ñ∈ {3, 4}，强调它们的对称性破坏。我们还表明，在适当的功能空间中没有达到最佳常数。