Inequalities for spherically symmetric solutions of the wave equation (Q1894822)

If \(f(x)\) is a radial function in \(L^p (\mathbb{R}^d)\) and \(2 \leq p < 2d/(d - 1)\), then the solution \(u(t,x)\) of the wave equation with the initial data \(u(0,x) = f(x)\), \(u_t (0,x) = 0\), \(x \in \mathbb{R}^d\), is an element of \(L^p (\mathbb{R}^d)\) for almost every \(t \in [-T,T]\) and its \(L^p\)-norm in \([-T,T] \times \mathbb{R}^d\) is bounded by the \(L^p\)-norm of \(f(x)\). The result is not true for general functions \(f\).