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5.13.1 The $1$-dimensional case600FCCB417C9

Let $\Omega $ be a domain (a connected open subset) in $\mathbb {C}$.

Definition 5.13.2.C3781B231934 A subharmonic function on $\Omega $ is a function $\varphi \colon \Omega \rightarrow [-\infty ,\infty )$ satisfying the following three conditions:

  1. 06D657B81863

    $\varphi \not\equiv -\infty $;

  2. 2BAC570CEF93

    $\varphi $ is upper semicontinuous;

  3. 249F615C259B

    $\varphi $ satisfies the sub-mean value inequality: For any $a\in \Omega $ and $r>0$ such that $\overline{B_1(a,r)}\subseteq \Omega $, we have

    \[ \varphi (a)\leq \frac{1}{2\pi }\int _0^{2\pi } \varphi (a+r\mathrm{e}^{\mathrm{i}\theta })\, \mathrm{d}\theta .{}^{\href{#sgpt-a0000001456}{1}} \]

We will denote the set of subharmonic functions on $\Omega $ by $\mathrm{SH}(\Omega )$.

Here, $B_1(a,r)\coloneqq \{ z\in \mathbb {C}:|z-a|<r\} $ denotes the open ball with center $a$ and radius $r$.

In fact, for each $a\in \Omega $, in 3, it suffices to require the sub-mean value inequality for all small enough $r>0$.

Intuitively, at a specific point $a\in \Omega $, Condition 2 says $\varphi (a)$ is no smaller than the upper limit of nearby values, while Condition 3 says $\varphi (a)$ is no larger than the mean. This intuition leads to the following rigidity theorem:

Theorem 5.13.3.FDD1E63B4457 Let $\varphi \colon \Omega \rightarrow [-\infty ,\infty )$ be a measurable function. Then the following are equivalent:

  1. E03504FC9B0D

    $\varphi $ is locally integrable and $\Delta \varphi \geq 0$.

  2. D517D85A39B6

    $\varphi $ coincides almost everywhere with a subharmonic function $\psi $ on $\Omega $.

Moreover, the subharmonic function $\psi $ in 2 is unique.

Proof. Suppose first that $\varphi\in L^1_{\mathrm{loc}}(\Omega)$ and $\Delta\varphi\geq0$. Apply the distributional characterization of subharmonic functions to the distribution defined by $\varphi$. It gives a subharmonic representative $\psi$ of this distribution. Equality of the distributions defined by two locally integrable functions is equality almost everywhere, so $\varphi=\psi$ almost everywhere.

Conversely, a subharmonic function on the connected domain $\Omega$ is locally integrable and has positive distributional Laplacian by the same theorem. Replacing it on a null set changes neither its distribution nor its local integrability. Finally, two subharmonic representatives that agree almost everywhere agree at every point: their values are the limits of their averages over shrinking balls. This also proves uniqueness. Multiplying the usual planar Laplacian by a fixed positive normalization constant does not change positivity.

Here in Condition 1, $\Delta \varphi =2\mathrm{i}\partial \bar\partial \varphi $ is the Laplacian in the sense of currents. This is a special case of theorem 5.13.8 below.

This theorem gives a very useful way of constructing subharmonic functions.

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