Weighted Lp-potential theory on homogeneous groups

Abstract

Two independent generalizations of nonlinear potential theory [1-3] are combined. The first generalization [4] was made in response to boundary-value problems for degenerate elliptical equations [5, 6], and the second [7-10] was motivated by certain problems related to anisotropic spaces of differentiable functions [11-16] and investigations in the area of quasielliptical equations [17, 18]. The approach to nonlinear potential theory proposed in the present article comprises two generalizations, and a number of propositions stated here for weighted potential theory are novel, for example, Frostman-type theorems concerning the relation between the weighted capacity and weighted Hausdorff volume (measure). Strong Maz'ya-type capacity inequalities [19] and two-weighted bounds for general type potentials are introduced as applications of the theory in the general situation. In conclusion weight-theoretic analogues of the Choquet theorem and Kellog properties are formulated and applications of these analogues to degenerate elliptical equations are presented.