Proper $r$-harmonic functions on the Thurston geometries

Abstract

For any positive natural number $r\in {\mathbb{N}}^{+}$ we construct new explicit proper $r$-harmonic functions on the celebrated $3$-dimensional Thurston geometries $\Sol$, $\Nil$, $\SL2$, $\H^2\times\rn$ and $\s^2\times\rn$.