Math. Nachr. zyxwvutsrq 169 (1994) 59-67 Appell Type Transformation for the Kolmogorov Operator By MIROSLAV BRZEZINA zyxwvut ') of Liberec (Received August 19, 1993) a2 a a ax2 ay at Abstract. For the Kolmogorov operator zyxwvuts L =: - + zyxw x - - - we describe all transformations mapping solutions of the equation Lu = 0 into solutions. 0. Introduction The Laplace operator and the heat operator are well known examples for differential operators generating harmonic spaces in the sense of H. BAUER, see e.g. [2]. The Kolmogorov operator generates also a harmonic space. In zyxwv [5], [7] and [9], potential theory of the Kolmogorov operator was investigated. In particular, a mean value theorem was proved and its potential theoretical interpretation was discussed; furthermore, a maximum principle and the Wiener type test for regularity was established. Harnack inequality and a mean value theorem are given in [3] for operators of Kolmogorov type. In this note we investigate L-harmonic morphism in the sense of the definition given below. It is well known that the Kelvin transformation for Laplace operator in IR3 (up to compositions with similarities) is the only one which maps the class of harmonic functions into itself; see [6], cf. [4] for zyxwvu IR". a. For the heat operator H = d - - in IR"+l, the Appell transformation plays the same at r61e as the Kelvin transformation for the Laplace operator, see e.g. [l], (81. In [8], H. LEUTWILER has shown that the Appell transformation is essentially the only transformation mapping caloric functions (i.e., the solutions of the equation Hu = 0) into another. A similar result holds also for the Kolmogorov operator as will be proved in this note. ') Financial support by the Volkswagen Stiftung is gratefully acknowledged.