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Transactions of the American Mathematical Society

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The Martin kernel and infima of positive harmonic functions


Author: Zoran Vondraček
Journal: Trans. Amer. Math. Soc. 335 (1993), 547-557
MSC: Primary 31C35; Secondary 31B05, 35B05, 35J15
DOI: https://doi.org/10.1090/S0002-9947-1993-1104202-1
MathSciNet review: 1104202
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Abstract: Let $ D$ be a bounded Lipschitz domain in $ {{\mathbf{R}}^n}$ and let $ K(x,z)$, $ x \in D$, $ z \in \partial D$, be the Martin kernel based at $ {x_0} \in D$. For $ x,y \in D$, let $ k(x,y) = \inf \{ h(x):h\;$positive$ \;$harmonic$ \;$in$ \; D, h(y) = 1\} $. We show that the function $ k$ completely determines the family of positive harmonic functions on $ D$. Precisely, for every $ z \in \partial D$, $ {\lim _{y \to z}}k(x,y)/k({x_0},y) = K(x,z)$. The same result is true for second-order uniformly elliptic operators and Schrödinger operators.


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  • [Ahl] L. V. Ahlfors, Möbius transformation in several dimensions, Ordway Lectures in Mathematics, University of Minnesota, 1981. MR 725161 (84m:30028)
  • [Caf] L. Caffarelli, E. Fabes, S. Mortola, and S. Salsa, Boundary behaviour of nonnegative solutions of elliptic operators in divergence form, Indiana J. Math. 30 (1981), 621-640. MR 620271 (83c:35040)
  • [Chi] F. Chiarenza, E. Fabes, and N. Garofalo, Harnack's inequality for Schrödinger operators and continuity of solutions, Proc. Amer. Math. Soc. 98 (1986), 415-425. MR 857933 (88a:35037)
  • [Cra] M. Cranston, E. Fabes, and Z. Zhao, Conditional gauge and potential theory for Schrödinger operator, Trans. Amer. Math. Soc. 307 (1988), 171-194. MR 936811 (90a:60135)
  • [Doo] J. L. Doob, Classical potential theory and its probabilistic counterpart, Springer, New York and Heidelberg, 1984. MR 731258 (85k:31001)
  • [Hun] R. A. Hunt and R. L. Wheeden, Positive harmonic functions on Lipschitz domains, Trans. Amer. Math. Soc. 147 (1970), 507-527. MR 0274787 (43:547)
  • [Leu] H. Leutwiler, On a distance invariant under Möbius transformation in $ {{\mathbf{R}}^n}$, Ann. Acad. Sci. Fenn. Ser. A I Math. 12 (1987), 3-17. MR 877574 (88a:31009)
  • [Mar] R. S. Martin, Minimal positive harmonic functions, Trans. Amer. Math. Soc. 49 (1941), 137-172. MR 0003919 (2:292h)
  • [Rao] M. Rao, Brownian motion and classical potential theory, Lecture Notes Ser., vol. 667, Aarhus Univ., Aarhus, 1977. MR 0440718 (55:13589)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1104202-1
Keywords: Positive harmonic functions, Martin kernel
Article copyright: © Copyright 1993 American Mathematical Society

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