Weighted Strichartz estimate for the wave equation and low regularity solutions

In this work we study weighted Sobolev spaces in Rⁿ generated by the Lie algebra of vector fields (1 + |x|²)1/2 ∂x_i, j = l,n...n. Interpolation properties and Sobolev embeddings are obtained on the basis of a suitable localization in Rⁿ. As an application we derive weighted Lq estimates for the solution of the homogeneous wave equation. For the inhomogeneous wave equation we generalize the weighted Strichartz estimate established in [5] and estab lish global existence result for the supercritical semilinear wave equation with non compact small initial data in these weighted Sobolev spaces.