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

TL;DR

This paper establishes a superposition principle for Riesz potentials on Lie groups of Heisenberg type, showing they are either p-subharmonic or p-superharmonic depending on parameters, extending previous results to a broader class of groups.

Contribution

It proves that Riesz potentials on Lie groups of Heisenberg type are necessarily p-subharmonic or p-superharmonic, generalizing recent superposition results to nonabelian stratified Lie groups.

Findings

Riesz potentials are either p-subharmonic or p-superharmonic on these groups.

The result extends superposition principles to a wide class of nonabelian stratified Lie groups.

The work generalizes previous results from abelian to nonabelian settings.

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 $$ R_\alpha(\rho)(g) = \int_{\G} N(g^{-1} g')^{\alpha-Q} \rho(g') dg', \qquad 0<\alpha<Q, $$ of a nonnegative function $\rho\in C_0(\G)$ 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.