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# The Martin Kernel and Infima of Positive Harmonic Functions

Zoran Vondraček
Transactions of the American Mathematical Society
Vol. 335, No. 2 (Feb., 1993), pp. 547-557
DOI: 10.2307/2154393
Stable URL: http://www.jstor.org/stable/2154393
Page Count: 11
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## Abstract

Let D be a bounded Lipschitz domain in Rn and let K(x, z), x ∊ D, z ∊ ∂D, be the Martin kernel based at $x_{0}\in D$. For x, y ∊ 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 ∊ ∂D, ${\rm lim}_{y\rightarrow z}k(x,y)/k(x_{0},y)=K(x,z)$. The same result is true for second-order uniformly elliptic operators and Schrödinger operators.

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