# Extension of subharmonic function: can someone explain the details?

In this paper we have the following situation on page 60. $$E$$ is a compact subset of $$\mathbb{R}^\tau\cup\{\infty\}$$ (one point compactification) for $$\tau\geq2$$, $$M_0$$ is a point in the boundary of $$E$$ and $$u$$ is subharmonic on an open neighborhood $$V$$ of $$M_0$$. M. Brelot says that we can extend $$u$$ to a function that is subharmonic on a neighborhood of$$E$$, that he steel calls $$u$$, by "modifying properly $$u$$ outside $$V$$". Can someone explains the detail of this extension?

By the way a function $$v$$ on an open set $$W$$ of $$\mathbb{R}^\tau\cup\{\infty\}$$ is called subharmonic if: either $$W$$ does not contain $$\infty$$ and in this case $$v$$ is subharmonic on $$W$$ in the usual sense, or $$W$$ contains $$\infty$$ and then 1) $$v$$ is upper semicontinuous at $$\infty$$ and 2) $$v(\infty)\leq$$ the mean integral value of $$u$$ over any ball $$B$$ that is relatively compact in $$W$$.

Recall that the extension of a subharmonic function to $$\mathbb{R}^\tau$$ is classic. If $$u$$ is subharmonic on a neighborhood of $$\overline{V}$$ (closure), where $$V$$ is a bounded open set in $$\mathbb{R}^\tau$$ such that $$\mathbb{R}^\tau\setminus\overline{V}$$ is connected, then there exists a subharmonic function $$\bar{u}$$ on $$\mathbb{R}^\tau$$ that coincides with $$u$$ on $$\overline{V}$$ (see Armitage and Gardiner, "classical potential theory", pg192).

But Brelot extends $$u$$ to an open set of $$\mathbb{R}^\tau\cup\{\infty\}$$ containg $$E$$. How??

• If $E$ does not contain $\infty$, you know how to extend to $R^n$. If $E$ contains $\infty$, suppose that $x_0\not\in E$, and perform the Kelvin transform which sends $x_0$ to $\infty$. If $E=R^n\cup\{\infty\}$ then there is no boundary point and no argument is needed. – Alexandre Eremenko Feb 23 at 12:37
• Consider the case of $E$ containing $\infty$. Suppose the ball $B(x_0,r)\subset \complement E\cap V$ ($\complement E$=complement of $E$) and $u^*$ is the Kelvin transfrom of $u$ relative to the sphere $\partial B(x_0,r)$. If I am following correctely, $u^*$ is subharmonic on a neighborhood of $E$, but the restriction of $u^*$ to $V$ does not coincide with $u$. I need the restriction of the extension to coincide with $u$ on $V$ or $\overline{V}\cap E$ (assume $u$ is subharmonic on a neighborhood of $\overline{V}$) . – M. Rahmat Feb 23 at 21:50
• If I understand correctly (unfortunately I do not speak French), Brelot refers here to one of his previous papers: Sur le rôle du point à l'infini dans la théorie des fonctions harmoniques, Ann. Éc. Norm. sup., t. 61, 1944. I did not follow this reference, but I guess the argument is not much different from the classical one. – Mateusz Kwaśnicki Feb 26 at 8:13
• Yes, he refers to part 20, "potential representation", of the paper you cited. But I only found there a generalization of the classic result that a subharmonic function can be locally represented as the sum of a potential and a harmonic function. I didn't get what does it have to do with the extension of subharmonic functions... – M. Rahmat Feb 27 at 0:52