Properties and some Applications of the Radial Integration Operators.

Radial integration operators I^α in bounded star–shaped domains are used in mathematical physics to find solution of biharmonic equation, proving of S.L. Sobolev embedding theorems, in the theory of special functions, etc. In this paper, we introduce the new operator J^α acting on functions defined in complements Ω of a star–shaped domains. By means of these operators three–dimensional quaternionic Kolosov–Muskhelishvili formulae in Ω is obtained. As application, the problem of an equilibrium of three dimensional elastic space with a ball cavity is solved in a closed form. [ABSTRACT FROM AUTHOR]