It is well known that, for space dimension \(n> 3\), one cannot generally expect \(L^1\)–\(L^p\) estimates for the solution of $$\begin{aligned} u_{tt}-\varDelta u = 0, \quad u(0,x)=0,\quad u_t(0,x)=g(x), \end{aligned}$$ where \((t,x)\in {\mathbb {R}}_{+}\times {{\mathbb {R}}^{n}}\). In this paper, we investigate the benefits in the range of \(1\le p \le q\) such that \(L^p\)–\(L^q\) estimates hold under the assumption of radial initial data. In the particular case of odd space dimension, we prove \(L^1\)–\(L^q\) estimates for \(1\le q \sigma _c(n)\), where the critical exponent \(\sigma _c(n)\) is the Strauss index.