It is well known that, for space dimension $$n> 3$$ n > 3 , one cannot generally expect $$L^1$$ L 1 – $$L^p$$ L p estimates for the solution of $$\begin{aligned} u_{tt}-\varDelta u = 0, \quad u(0,x)=0,\quad u_t(0,x)=g(x), \end{aligned}$$ u t t - Δ u = 0 , u ( 0 , x ) = 0 , u t ( 0 , x ) = g ( x ) , where $$(t,x)\in {\mathbb {R}}_{+}\times {{\mathbb {R}}^{n}}$$ ( t , x ) ∈ R + × R n . In this paper, we investigate the benefits in the range of $$1\le p \le q$$ 1 ≤ p ≤ q such that $$L^p$$ L p – $$L^q$$ L q estimates hold under the assumption of radial initial data. In the particular case of odd space dimension, we prove $$L^1$$ L 1 – $$L^q$$ L q estimates for $$1\le q <\frac{2n}{n-1}$$ 1 ≤ q < 2 n n - 1 and apply these estimates to study the global existence of small data solutions to the semilinear wave equation with power nonlinearity $$ |u|^{\sigma }$$ | u | σ , $$\sigma >\sigma _c(n)$$ σ > σ c ( n ) , where the critical exponent $$\sigma _c(n)$$ σ c ( n ) is the Strauss index.