Abstract
LetL be a second order elliptic differential operator on a differentiable manifoldM and let 1 <α≤2. We investigate connections bewween classU of all positive solutions of the equationLu=u α and classH of all positiveL-harmonic functions (i.e., solutions of the equationsLh=0). Putu∈U 0 ifu∈U and ifu≤h for someh∈H. To everyu∈U 0 there corresponds the minimalL-harmonic functionh u which dominatesu andu→h u is a 1–1 mapping fromU 0 onto a subsetH 0 ofH. The inverse mapping associates with everyh∈H 0 the maximal element ofU dominated byh.
Supposeg(x, dy) is Green's kernel,k(x, y) is the Martin kernel and ϖM is the Martin boundary associated withL. A subset Γ of ϖM is calledR-polar if it is not hit by the rangeR of the (L, α)-superdiffusion. It is calledM-polar if\(\int\limits_M {g\left( {c,dx} \right)[\int\limits_\Gamma {k(x,y)v(dy)]^\alpha } } \) is equal to 0 or ∞ for everyc∈M and every measure ρ.
Everyh∈H has a unique representation\(h(x) = \int\limits_{\partial M} {k\left( {x,y} \right)v\left( {dy} \right)} \) where ρ is a measure concentrated on the minimal partM * of ϖM.
We show that the condition:
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(a)
ρ(Γ)=0 for allR sets Γ is necessary and the condition:
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(b)
ρ(Γ)=0 for allM-polar sets Γ is sufficient forh to belong toH 0. IfM is a bounded domain of classC 2, λ in ℝd, then conditions (a) and (b) are equivalent and therefore each of them characterizesH 0. This was conjectured by Dynkin a few years ago and proved in a recent paper of Le Gall forL=Δ, α=2 and domains of classC 5.
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Partially supported by National Science Foundation Grant DMS-9301315.
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Dynkin, E.B., Kuznetsov, S.E. Solutions ofLu=u α dominated byL-harmonic functions. J. Anal. Math. 68, 15–37 (1996). https://doi.org/10.1007/BF02790202
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DOI: https://doi.org/10.1007/BF02790202