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