In this work we study weighted Sobolev spaces in $\mathbf{R}^{n}$
generated by the Lie algebra of vector fields
\[
\left(1+\mid x\mid^{2}\right)^{1/2}\partial_{x_{j}},\; j=1,...,n.
\]
Interpolation properties and Sobolev embeddings are obtained on the
basis of a suitable localization in $\mathbf{R}^{n}$. As an application
we derive weighted L$^{q}$ estimates for the solution of the homogeneous
wave equation. For the inhomogeneous wave equation we generalize the
weighted Strichartz estimate established in $\left[5\right]$ and
establish global existence result for the supercritical semilinear
wave equation with non compact small initial data in these weighted
Sobolev spaces.
