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By adapting the mass transportation technique of Cordero-Erausquin, Nazaretand 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 workwith the fractional dimension $n_a = n+a$. The extremal functions in theweighted Sobolev inequalities are fully characterized. Using a dimensionreduction argument and the weighted Sobolev inequalities, we can reproduce asubfamily of the sharp GN inequalities on the Euclidean space due to Del Pinoand Dolbeault, and obtain some new sharp GN inequalities as well. Our weightedinequalities 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.

Sharp weighted Sobolev and Gagliardo-Nirenberg inequalities on half space via mass transport and consequences