Norm inequalities for spherical means (Q1061335)

For \(f\in L^ 1_{loc}({\mathbb{R}}^ n)\) set \[ F_ x(t)=\int_{s^{n- 1}}f(x-ty) d\sigma (y),\quad x\in {\mathbb{R}},\quad t\in {\mathbb{R}}, \] where \(\sigma\) denotes the surface measure on \(S^{n-1}\). Sharp results on the regularity of the function \(F_ x\) are given when \(f\in L^ p({\mathbb{R}}^ n)\), \(n\geq 2\). The results extend an earlier theorem of the author in ibid. 96, 277-291 (1983; Zbl 0519.42018).