Hardy’s inequality in higher dimension

Thm: Supppose {u\in H^1(\mathbb{R}^n)} with {n\geq 3} prove that

\displaystyle \int_{\mathbb{R}^n}\frac{u^2}{|x|^2}dx\leq C(n)\int_{\mathbb{R}^n}|\nabla u|^2dx

Proof: Suppose {u\in C^\infty_c(\mathbb{R}^n)} firstly. We can find ball {B_R(0)} large enough such that {Supp. u\subset B_R}, then we only need to prove the inequality within {B_R}.

Let

\displaystyle I=\int_{B_R}\frac{u^2}{|x|^2}dx=\frac{-1}{2}\int_{B_R}x\cdot D\left(\frac{1}{|x|^2}\right)u^2dx

Apply the Green identity on {B_R\backslash B_\epsilon(0)},

\displaystyle \int_{B_R\backslash B_\epsilon(0)}x\cdot D\left(\frac{1}{|x|^2}\right)u^2dx+\int_{B_R\backslash B_\epsilon(0)}\frac{1}{|x|^2}D(xu^2)dx=\int_{\partial B_\epsilon}\frac{1}{|x|^2}xu^2\cdot\nu dx

where {\nu} is the unit normal vector of {\partial B_\epsilon}

\displaystyle \left|\int_{\partial B_\epsilon}\frac{1}{|x|^2}xu^2\nu dx\right|\leq \int_{B_\epsilon}\frac{u^2}{|x|^2}dx\leq \frac{1}{\epsilon}\leq \frac{1}{\epsilon}\sup u^2n w_n\epsilon^{n-1}

Let {\epsilon\rightarrow 0}, we have

\displaystyle -2I+\int_{B_R}\frac{1}{|x|^2}D(xu^2)dx=0

\displaystyle -2I+\int_{B_R}\frac{nu^2}{|x|^2}+\frac{1}{|x|^2}2uDu\cdot x=0

\displaystyle -2I+nI+2\int_{B_R}\frac{u}{|x|^2}Du\cdot xdx=0

\displaystyle I=\frac{2}{2-n}\int_{B_R}\frac{u}{|x|^2}Du\cdot xdx

From Holder inequality, we have

\displaystyle I\leq \left(\frac{2}{n-2}\right)^2\int_{B_R}|\nabla u|^2dx

Next suppose {u\in H^1(\mathbb{R}^n)}. Since {C^\infty_c(\mathbb{R}^n)} is dense in {H^1(\mathbb{R}^n)}, there exist {u_k\subset C^\infty_c(\mathbb{R}^n)\rightarrow u} in {H^1(\mathbb{R}^n)}. From Fatou’s lemma

\displaystyle \int_{\mathbb{R}^n}\frac{u^2}{|x|^2}dx\leq \lim\inf_{k\rightarrow \infty}\int_{\mathbb{R}^n}\frac{u_k^2}{|x|^2}dx\leq C(n)\lim\inf_{n\rightarrow \infty}\int_{B}|Du_k|^2=C(n)\int_{\mathbb{R}^n}|\nabla u|^2dx

Remark: If {n=2}, we should prove that if {u\in C^\infty_c(\Omega)}, where {\Omega=\{x|1\leq |x|<\infty\}}

\displaystyle \int_{\Omega}\frac{u^2}{|x|^2\ln^2|x|}dx\leq 4\int_{\Omega}|Du|^2dx

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