Potential theory on homogeneous groups. (Russian) Zbl 0673.31005

In the present article \(L_ p\)-theory of potential on any homogeneous group [see G. B. Folland and E. M. Stein, Hardy spaces on homogeneous groups (1982; Zbl 0508.42025)] with respect to kernels which are functions of homogeneous norm is developed. In the linear case \((p=2)\) the classical results of potential theory are generalized: generalized maximum principle, theorems of Evans-Vasilesco, Frostman, Nevanlinna, Ohtsuka, Kellogg’s property [See N. S. Landkof, Foundations of the modern potential theory (1972; Zbl 0253.31001)]. In the nonlinear case (p\(\neq 2)\) the results of V. G. Maz’ya and V. P. Khavin, D. R. Adams and N. Meyers, L. I. Hedberg and T. H. Wolff are generalized.
The author gives references to own papers where one can find the applications of the above-mentioned results to nonlinear quasi-elliptic equations. Contents: 1. Homogeneous groups. Examples. 2. Energy and nonlinear potentials. Volf’s inequality. 3. Theorem of the Evans- Vasileco’s type. 4. Comparison of the capacity and metric characteristics of the sets. 5. Thin sets in potential theory.
Reviewer: A.D.Bendikov


31C05 Harmonic, subharmonic, superharmonic functions on other spaces
31C15 Potentials and capacities on other spaces
31C12 Potential theory on Riemannian manifolds and other spaces
22E30 Analysis on real and complex Lie groups
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