  There are at least two directions concerning the extension of classical sharp
Hardy-Littlewood-Sobolev inequality: (1) Extending the sharp inequality on
general manifolds; (2) Extending it for the negative exponent
$\lambda=n-\alpha$ (that is for the case of $\alpha>n$). In this paper we
confirm the possibility for the extension along the first direction by
establishing the sharp Hardy-Littlewood-Sobolev inequality on the upper half
space (which is conformally equivalent to a ball). The existences of extremal
functions are obtained; And for certain range of the exponent, we classify all
extremal functions via the method of moving sphere.
