Poincare's inequality and Sobolev spaces with monomial weights

In this paper, we use a weighted version of Poincare's inequality to study density and extension properties of weighted Sobolev spaces over some open set & omega;& SUBE;RN$\Omega \subseteq \mathbb {R}<^>N$. Additionally, we study the specific case of monomial weights w(x1, horizontal ellipsis ,xN)=∏i=1Nxiai,ai & GE;0$w(x_1,\ldots ,x_N)=\prod _{i=1}<^>N\left|x_i \right|<^>{a_i},\ a_i\ge 0$, showing the validity of a weighted Poincare inequality together with some embedding properties of the associated weighed Sobolev spaces.