Estimates for the Products of the Green Function and the Martin Kernel

Let Ω be a proper subdomain of ℝn, n ≥ 2, and let x0∈ Ω be fixed. By GΩ and KΩ we denote the Green function and the Martin kernel for Ω, respectively. Under a certain assumption on Ω near a boundary point ξ, we show that the product GΩ(x, x0)KΩ(x, ξ) is comparable to |x - ξ|2-n for x in a nontangential cone with vertex at ξ. We also give an estimate for the product KΩ(x, ξ)KΩ(x,η) in a uniform domain, where η is another boundary point.

BOUNDARY BEHAVIOUR OF QUOTIENTS OF MARTIN KERNELS

Given two intersecting domains, we investigate the boundary behaviour of the quotient of Martin kernels of each domain. To this end, we give a characterization of minimal thinness for a difference of two subdomains in terms of Martin kernels of each domain. As a consequence of our main theorem, we obtain the boundary growth of the Martin kernel of a Lipschitz domain, which corresponds to earlier results for the boundary decay of the Green function for a Lipschitz domain investigated by Burdzy, Carroll and Gardiner.