  By adapting the mass transportation technique of Cordero-Erausquin, Nazaret
and Villani, we obtain a family of sharp Sobolev and Gagliardo-Nirenberg (GN)
inequalities on the half space $\mathbf{R}^{n-1}\times\mathbf{R}_+$, $n\geq 1$
equipped with the weight $\omega(x) = x_n^a$, $a\geq 0$. It amounts to work
with the fractional dimension $n_a = n+a$. The extremal functions in the
weighted Sobolev inequalities are fully characterized. Using a dimension
reduction argument and the weighted Sobolev inequalities, we can reproduce a
subfamily of the sharp GN inequalities on the Euclidean space due to Del Pino
and Dolbeault, and obtain some new sharp GN inequalities as well. Our weighted
inequalities are also extended to the domain $\mathbf{R}^{n-m}\times
\mathbf{R}^m_+$ and the weights are $\omega(x,t) = t_1^{a_1}\dots t_m^{a_m}$,
where $n\geq m$, $m\geq 0$ and $a_1,\cdots,a_m\geq 0$. A weighted
$L^p$-logarithmic Sobolev inequality is derived from these inequalities.
