The inversion of fractional integrals on a sphere (Q1802752)

The author considers the Riesz type potential \[ f(x)=c_{n,\alpha} \int_{\Sigma^ n} | x-y|^{\alpha-n} \varphi(y)dy, \qquad \alpha>0,\;x\in\Sigma^ n, \] over the \(n\)-dimensional sphere \(\Sigma^ n\) in \(R^{n+1}\), \(dy=ds(y)\) being the surface element, with the standard modification int he case \(\alpha=n+2k\), \(k=0,1,\dots\;\). He considers the problem of finding \(\varphi(y)\) by a given \(f(x)\). This problem was solved in the case \(0<\alpha<2\) by \textit{P. M. Pavlov} and the reviewer [Dokl. Akad. Nauk SSSR 276, 546-550 (1984); Transl. in Soviet Math. Dokl. 29, 549-553 (1984; Zbl 0594.46026)] with the inverse operator constructed as a spherical hypersingular operator. The author applies the representation of \(f(x)\) in terms of mean values over planar sections, see reviewer's paper [Trudy Mat. Inst. Steklova 156, 157-222 (1980; Zbl 0448.42012); Transl. in Proc. Stekov Inst. Math. 156, 173-243 (1983; Zbl 0518.42025)] and uses the known idea of the Marchaud difference regularization of one-dimensional divergent integrals. In this way he arrives at the construction which is an inversion operator for all \(\alpha>0\). The Liouville type space \(L_ p^ \alpha(\Sigma^ n)\) of functions on \(\Sigma^ n\) with the fractional smoothness is characterized in terms connected with the constructed inverse operator. Finally, in the case \(\alpha=n+2k\) it is shown that the considered potential operator is, in general, a Noetherian operator in the space \(L_ p(\Sigma^ n)\) and its \(d\)-characteristic depends on the radius of the sphere.