Weighted Strichartz Estimate for the Wave Equation and Low Regularity Solutions

Abstract

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.