**Hardy inequality** Suppose that $u$ belongs to Schwartz space $\mathcal{S}(\mathbb{R}^n)$ with $n\geq 3$, then we have $$\int \f{\abs{u}^2}{\abs{x}^2} dx \leq C\int \abs{\grad u}^2 dx.$$ **//Proof//** Using that $\abs{ \grad u + \lambda \f{x}{\abs{x}^2} u }^2 \geq 0$ for all $\lambda\in\mathbb{R}$，and then expanding it and integrating by parts for the mixed term, we will obtain the required inequality by choosing $\lambda=(2-n)/2$. The best constant $C=(2/(n-2))^2$. **Remark** See [SSW03] for a general Sobolev-Hardy inequality and see [Fra11] for a physical relevant application (or explanation). More relevant resources are [[http://ima.ucv.cl/rmenares/Recherche/Otros/inequalities.pdf|THREE VARIANTS OF THE HARDY INEQUALITY]], [[http://ricerca.mat.uniroma3.it/AnalisiNonLineare/preprints/BadialeTarantelloxx.pdf|A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics]], [[http://www.mai.liu.se/~vlmaz/pdf/maz9.pdf|Critical Hardy–Sobolev Inequalities]]. === References [SSW03] S. Secchi, D. Smets, M. Willemc, [[http://arxiv.org/pdf/math/0212083v3.pdf|Remarks on a Hardy-Sobolev inequality]] [Fra11] R. L. FRANK, [[https://web.math.princeton.edu/~rlfrank/sobweb1.pdf|SOBOLEV INEQUALITIES AND UNCERTAINTY PRINCIPLES IN MATHEMATICAL PHYSICS]]