## Hardy’s inequality in higher dimension

**Thm: **Supppose with prove that

**Proof: **Suppose firstly. We can find ball large enough such that , then we only need to prove the inequality within .

Let

Apply the Green identity on ,

where is the unit normal vector of

Let , we have

From Holder inequality, we have

Next suppose . Since is dense in , there exist in . From Fatou’s lemma

**Remark:** If , we should prove that if , where

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