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 Schrodinger equation Duðx Þ¼ QðxÞuðxÞ, QðxÞb 0o nT; polysuperharmonic functions and poly- potentials on T are defined and a Riesz-Martin representation for positive polysuper- harmonic functions is obtained. In this note, we study some classical potential-theoretic concepts like balayage, domination principle etc. in the context of a tree T and introduce the notions of polysuperharmonic functions and polypotentials on T and obtain some of their properties. The tree T is taken in the sense of Cartier's (4), a graph with infinite vertices, connected, locally finite and no circuits, provided with a transition probability structure. Bajunaid et al. (1) show that the har- monic functions on the vertices of T can be linearly extended to the edges, so that the extended functions verify the axioms 1, 2, 3 of Brelot. Consequently, some of the properties of harmonic functions and potentials on T can be im- mediately deduced from the axiomatic potential theory. However, on many occasions, direct proofs of theorems about harmonic functions on T are simpler and give more informations in comparison to those deduced from the axiomatic theory. Secondly, some theorems in the axiomatic theory require more assumptions than the axioms 1, 2, 3 only. One such is the converse to the Riesz representation theorem in a harmonic space W which states that given a positive Radon measure m on an open set o in W, there exists a superharmonic function s on o with associated measure m in a local Riesz representation. To prove this, we need the axiom of analyticity (de La Pradelle (6)) which is not generally valid on T. However, this converse to the Riesz representation is true on T (Theorem 2.4). Thirdly, for polyharmonic