## Hardy inequality in dimesion 2

Suppose is a smooth function defined in in the plane , assume on and also has compact support, then

There is a way presented by my advisor to explain why the strange function pops up here. First we transform LHS to polar coordinates

which leads us to start with a very general

Suppose , then

So we only need to find and such that

Actually one can solve the ODE

to get , . Plugging in this function back to the above proof gives you the desired inequality.

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