The Martin kernel and infima of positive harmonic functions
HTML articles powered by AMS MathViewer
- by Zoran Vondraček PDF
- Trans. Amer. Math. Soc. 335 (1993), 547-557 Request permission
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 $k(x,y) = \inf \{ h(x):h\;\text {positive}\;\text {harmonic}\;\text {in}\; D, h(y) = 1\}$. 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.References
- Lars V. Ahlfors, Möbius transformations in several dimensions, Ordway Professorship Lectures in Mathematics, University of Minnesota, School of Mathematics, Minneapolis, Minn., 1981. MR 725161
- L. Caffarelli, E. Fabes, S. Mortola, and S. Salsa, Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana Univ. Math. J. 30 (1981), no. 4, 621–640. MR 620271, DOI 10.1512/iumj.1981.30.30049
- F. Chiarenza, E. Fabes, and N. Garofalo, Harnack’s inequality for Schrödinger operators and the continuity of solutions, Proc. Amer. Math. Soc. 98 (1986), no. 3, 415–425. MR 857933, DOI 10.1090/S0002-9939-1986-0857933-4
- M. Cranston, E. Fabes, and Z. Zhao, Conditional gauge and potential theory for the Schrödinger operator, Trans. Amer. Math. Soc. 307 (1988), no. 1, 171–194. MR 936811, DOI 10.1090/S0002-9947-1988-0936811-2
- J. L. Doob, Classical potential theory and its probabilistic counterpart, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 262, Springer-Verlag, New York, 1984. MR 731258, DOI 10.1007/978-1-4612-5208-5
- Richard A. Hunt and Richard L. Wheeden, Positive harmonic functions on Lipschitz domains, Trans. Amer. Math. Soc. 147 (1970), 507–527. MR 274787, DOI 10.1090/S0002-9947-1970-0274787-0
- Heinz Leutwiler, On a distance invariant under Möbius transformations in $\textbf {R}^n$, Ann. Acad. Sci. Fenn. Ser. A I Math. 12 (1987), no. 1, 3–17. MR 877574, DOI 10.5186/aasfm.1987.1219
- Robert S. Martin, Minimal positive harmonic functions, Trans. Amer. Math. Soc. 49 (1941), 137–172. MR 3919, DOI 10.1090/S0002-9947-1941-0003919-6
- Murali Rao, Brownian motion and classical potential theory, Lecture Notes Series, No. 47, Aarhus Universitet, Matematisk Institut, Aarhus, 1977. MR 0440718
Additional Information
- © Copyright 1993 American Mathematical Society
- 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