Riesz potentials and p-superharmonic functions in Lie groups of Heisenberg type

Abstract

We prove a superposition principle for Riesz potentials of nonnegative continuous functions on Lie groups of Heisenberg type. More precisely, we show that the Riesz potential $$ R_α(ρ)(g) = \int_{\G} N(g^{-1} g')^{α-Q} ρ(g') dg', \qquad 0<α<Q, $$ of a nonnegative function $ρ\in C_0(\G)$ on a group $\G$ of Heisenberg type is necessarily either $p$-subharmonic or $p$-superharmonic, depending on $p$ and $\alpha$. Here $N$ denotes the non-isotropic homogeneous norm on such groups, as introduced by Kaplan. This result extends to a wide class of nonabelian stratified Lie groups a recent remarkable superposition result of Lindqvist and Manfredi.

Cited by in corpus (1)

• Stability theorems for H-type Carnot groups