Estimates for radial solutions to the wave equation

Abstract:

We consider the Cauchy problem for the wave equation with null initial position and radial initial velocity $\psi$. With the solution $u$ of this problem we define the operator $S(\psi )=\sup _{t>0}t^{-1}|u(x,t)|$. Using simple one-dimensional operators to bound pointwise the operator $S$ we obtain weighted $L^p$ estimates with power weights. Even for the unweighted estimates the result for dimension higher than 3 differs from the one with general functions.