The paper is devoted to the analysis of Riesz potentials \[ (K^{\alpha}\!_{\phi})(x)=C_{n,\alpha}\int_{B_{\alpha}\!^{(n)}} \phi(| y|)dy/| x-y|^{n-\alpha},\quad 0<\alpha<n \] where \(C_{n,\alpha}=(2^{\alpha}\pi^{n/2}\Gamma(\alpha /2))^{- 1}\Gamma((n-\alpha)/2)\), \(B_{\alpha}\!^{(n)}\) is a sphere of radius \(\alpha\leq \infty\) in \(R^ n\) with centre in the origin, \(\phi\) (\(| y|)\) belongs to the weighted space \(L_ p(B_{\alpha}\!^{(n)},| x|^{\nu}), 1<p<\infty\), \(n\geq 1\). The author gives new representations of functions and illustrates some global estimations.