# Reference for a theorem on subharmonic functions

I need a reference for a theorem that states: Let $$D$$ be a domain of $$\mathbb{R}^{m}$$ and let $$K(x)= \log|x|$$ if $$m=2$$, and $$K(x)=|x|^{2-m}$$ if $$m>2$$. Let $$u$$ be a subharmonic function on $$D$$. Then there is a unique Borel measure $$\mu$$ such that for all compact $$E$$ in $$D$$ we have $$u(x)=\int_{\partial E}u(\zeta)d\omega_{x}(\zeta)-\int_{E}K(x-\zeta)d\mu(\zeta),$$ where $$\omega_{x}$$ is the harmonic measure at $$x$$.

In Hayman and Kenney's book ("subharmonic functions", vol. 1 Theorem 3.9, pg 104) this theorem is given but instead of the first integral, they give a function $$h$$ harmonic in the interior of $$E$$ only. Also, on pg 120, the explicit form of the first integral is given but the integrand of the second integral is the green function $$g$$.

Same things in Armitage and Gardiner's book ("Classical potential theory"). I would appreciate if someone could give me a reference or explain how to go from one form to the other.

• You need some assumptions on $E$ to define $\omega_x(\zeta)$. And more assumptions to make your formula true. – Alexandre Eremenko Sep 13 '19 at 12:08

Perhaps I misunderstood the question: you are right, there are two expressions available for $$u$$.
1. We have $$u(x) = \int_{\partial E} u(z) \omega_E^x(dz) - \int_E G_E(x, y) \mu(dy) ,$$ where $$\omega^E_x$$ is the harmonic measure, $$G_E$$ is the Green's function, and $$\mu = -\Delta u$$ in the sense of distributions.
2. We also have $$u(x) = h(x) - \int_E K(y - x) \mu(dy) ,$$ where $$K$$ is the Newtonian kernel (the Green's function for $$\mathbb{R}^m$$) and $$\mu$$ is as above.
These two expressions are of course closely related: since $$G_E(x, y) = K(y - x) - \int_{\partial E} K(z - y) \omega^x_E(dz) ,$$ we have $$h(x) = \int_{\partial E} \biggl( u(z) + \int_E K(z - y) \mu(dy) \biggr) \omega^x_E(dz) .$$
• Thanks. Is the first expression with Green's function formally valid if $E$ is a compact with empty interior? – M. Rahmat Sep 13 '19 at 9:18
• I think so (although I am used to thinking $E$ is an open set). The probabilistic argument works for general sets: if $T_E$ is the first exit time, then $G_E$ is the density function of the mean occupation measure, $\omega_E$ is the distribution at $T_E$, and the desired expression is a consequence of the strong Markov property of Brownian motion. – Mateusz Kwaśnicki Sep 13 '19 at 9:26