Weighted Decay Estimates for the Wave Equation

In this work we study weighted Sobolev spaces in Rn generated by the Lie algebra of vector fields (1+x2)1/2 ∂xj,j=1, …, 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 by V. Georgiev (1997, Amer. J. Math.119, 1291–1319) and establish global existence results for the supercritical semilinear wave equation with non-compact small initial data in these weighted Sobolev spaces.