On non-commutative extensions of $\widehat{\G}_a$ by $\widehat{\Gg}^{(M)}$\\ over an ${\F}_p$-algebra

Abstract

Potential theory on a Cartier tree $T$ is developed on the lines of the classical and the axiomatic theories on harmonic spaces. The harmonic classifications of such trees are considered; the notion of a subordinate structure on $T$ is introduced to consider more generally the potential theory on $T$ associated with the Schrödinger equation $\Delta u\left( x\right) =Q\left(x\right) u\left( x\right) ,Q\left( x\right) \geq 0$ on $T$; polysuperharmonic functions and polypotentials on $T$ are defined and a Riesz-Martin representation for positive polysuperharmonic functions is obtained.