Weighted \(L_ p\) potential theory on homogeneous groups. (English. Russian original) Zbl 0743.31008

Vodop’yanov, S. K.

Sov. Math., Dokl. 42, No. 2, 265-270 (1991); translation from Dokl. Akad. Nauk SSSR 314, No. 1, 37-41 (1990).

A homogeneous group is by definition a connected, simply connected nilpotent Lie group \(G\) with a one-parameter group of dilations \(\exp(A \hbox{ln }t)\), \(t>0\), acting on its Lie algebra; \(A\) is a diagonalizable linear operator on the Lie algebra with positive eigenvalues. In this context, appropriate weight functions on \(G\), neither too large nor yet too small, are used to define and study the weighted energy of a measure \(\mu\) on \(G\) (which is the \(d\mu\)-integral over \(G\) of the weighted nonlinear potential of \(\mu\)), the weighted Riesz capacity of a set \(E\), the weighted Hausdorff measure of \(E\subset G\). The author announces quite a few inequalities involving these quantities as well as several properties of the space of weighted Riesz potentials. No proofs.

Reviewer: E.J.Akutowicz (Montpellier)

MSC:

31C12	Potential theory on Riemannian manifolds and other spaces
43A15	\(L^p\)-spaces and other function spaces on groups, semigroups, etc.

Keywords:

weighted nonlinear potential of a measure; homogeneous group; weight functions; weighted energy; weighted Riesz capacity; weighted Hausdorff measure