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**L\({}_ p\)-theory of potentials and quasi-conformal mappings on homogeneous groups.**
*(Russian)*
Zbl 0714.31005

Tr. Inst. Mat. 14, 45-89 (1989).

The theory of potentials on homogeneous groups is developed in the paper; both cases, linear and nonlinear, are generalized to this case. At first Riesz potentials and nonlinear potentials of a measure are defined and the term of energy is investigated. A capacity is defined and some capacity and metric characteristics of sets are compared. In the linear case \((p=2)\) some classical theorems of potential theory are generalized to the case of homogeneous groups - for example generalized maximum principle, theorem of Evans-Vasilesco, Frostman’s theorem. In the nonlinear case some results of Maz’ya and Khavin, Meyers, Adams, Hedberg and Volf are generalized. Also a case analogous to the Bessel kernels is investigated. Further function spaces on homogeneous groups and a capacity on those spaces are investigated. The last part of the paper is devoted to the function spaces and quasi-conformal mappings on homogeneous groups.

Reviewer: M.Dont

### MSC:

31C45 | Other generalizations (nonlinear potential theory, etc.) |

31C15 | Potentials and capacities on other spaces |