The Bochner-Riesz means of order \(\delta\geq 0\) for suitable test functions are defined via the Fourier transform by \((S_ R^ \delta f)\sphat (\xi)=(1-|\xi|^ 2 R^ 2)_ +^ \delta\widehat f(\xi)\). Let \(\delta(p,n)= n/p-(n+1)/2\): the critical index. S. Chanillo and B. Muckenhoupt have proved that \(S_ R^{\delta(p,n)}f\) of radial functions in \(L^ p(\mathbb{R}^ n)\) are in weak-\(L^ p(\mathbb{R}^ n)\), i.e., in the Lorentz space \(L^{p,\infty}(\mathbb{R}^ n)\), \(1\leq p0}| S_ R^{\delta(p,n)}f(x)|\) is bounded on the subspace of radial functions in \(L^{p,\infty}(\mathbb{R}^ n)\). Hence, if \(f\) is radial and in the \(L^{p,\infty}(\mathbb{R}^ n)\) closure of test functions, \(S_ R^{\delta(p,n)}f(x)\) converges, as \(R\to\infty\), to \(f(x)\) for almost every \(x\in\mathbb{R}^ n\) and in the norm of \(L^{p,\infty}(\mathbb{R}^ n)\). They show however that the \(S_ R^{\delta(p,n)}f(x)\) for \(f=| x|^{-n/p}\), which belongs to \(L^{p,\infty}(\mathbb{R}^ n)\) but not to the closure of test functions, converges for no \(x\). For Fourier analysis of radial functions of \(\mathbb{R}^ n\) they treat the setting of Fourier-Bessel (Hankel) expansions in the Lorentz spaces \(L^{p,r}(\mathbb{R}_ +,x^{2\alpha+1}dx)\) \((\alpha>-1/2)\).