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In the present paper we shall establish n-dimensional Hardy's inequalitieswith non-doubling weight functions of the distance to the boundary, where theboundary is a $C^2$ class bounded domain of $R^N$. This work is essentiallybased on one dimensional weighted Hardy's inequalities with one-sided boundarycondition and sharp remainders. As weights we admit rather general ones thatmay vanish or blow up in infinite order such as $e^{-1/t}$ or $e^{1/t}$ at$t=0$ in one dimensional case.

Hardy's inequalities with non-doubling weights and sharp remainders