Basic Properties of Harmonic, Subharmonic, and Superharmonic Functions

Abstract

Let B = B(ξ₀, δ) and if ξ ∈ B denote by ξ′ the image of ξ under inversion in ∂B. That is, ξx′ is on the ray from ξ₀ through ξ, and |ξ – ξ₀| |ξx′ – ξ₀| = δ². To simplify the notation take ξ₀ = O. Then G _B , as defined by

$$ {G_B}\left\{ \begin{gathered} \log \frac{{|\zeta ' - \eta ||\zeta |}}{{\delta |\zeta - \eta |}} \left( { = \log \frac{\delta }{{|\eta |}} if \zeta = 0} \right) \hfill \\ for N = 2 \hfill \\ |\zeta - \eta {|^{{2 - N}}} - {\left( {\frac{\delta }{{|\zeta |}}} \right)^{{N - 2}}} \left( { = |\eta {|^{{2 - N}}} if \zeta = 0} \right) \hfill \\ for N > 2 \hfill \\ \end{gathered} \right. $$

(1.1)

with the understanding that G _B (ξ, ξ)= +∞, satisfies items (ix′)–(ivx′) of Section 1.8, so that harmonic measure for B is given by

$$ {\mu_B}(\zeta, d\eta ) = - \frac{1}{{\pi_N^{'}}}{D_{{{n_{\eta }}}}}{G_B}(\zeta, \eta ){l_{{N - 1}}}(d\eta ) = \frac{{K(\eta, \zeta )}}{{{\pi_N}{\delta^{{N - 1}}}}}{l_{{N - 1}}}(d\eta ) $$

(1.2)

where l _N–1 here refers to surface area on ∂B and

$$ K(\eta, ) = {\delta^{{N - 2}}}\frac{{{\delta^{{N - 2}}} - |\zeta {|^2}}}{{|\eta - \zeta {|^N}}} $$

(1.3)