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

Vodop’yanov, S. K.

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

Keywords:

potentials on homogeneous groups; Riesz potentials; nonlinear potentials; capacity; maximum principle; Frostman’s theorem; quasi-conformal mappings