Solutions of Lu=u(alpha) dominated by L-harmonic functions

Let L be a second order elliptic differential operator on a differentiable manifold M and let 1 < alpha less than or equal to 2. We investigate connections between class U of all positive solutions of the equation Lu = u(alpha) and class H of all positive L-harmonic functions (i.e., solutions of the equations Lh = 0). Put u is an element of U-0 if u is an element of U and if u less than or equal to h for some h is an element of H. To every u is an element of U-0 there corresponds the minimal L-harmonic function h(u) which dominates u and u --> h(u) is a 1-1 mapping from U-0 onto a subset H-0 of H. The inverse mapping associates with every h is an element of H-0 the maximal element of U dominated by h. Suppose g(x, dy) is Green's kernel, k(x, y) is the Martin kernel and partial derivative M is the Martin boundary associated with L. A subset Gamma of partial derivative M is called R-polar if [GRAPHICS] it is not hit by the range R of the (L, alpha)-superdiffusion. It is called M-polar if is equal to 0 or infinity for every c is an element of M and every measure nu. Every h is an element of H has a unique representation [GRAPHICS] where nu is a measure concentrated on the minimal part M* of partial derivative M. We show that the condition: (a) nu(Gamma) = 0 for all R-polar sets Gamma is necessary and the condition: (b) nu(Gamma) = 0 for all M-polar sets Gamma is sufficient for h to belong to H-0. If M is a bounded domain of class C-2,C-lambda in R(d), then conditions (a) and (b) are equivalent and therefore each of them characterizes H-0. This was conjectured by Dynkin a few years ago and proved in a recent paper of Le Gall for L = Delta, alpha = 2 and domains of class C-5.