Theorem 3.4.14 (Riesz decomposition).F04DBD614238 Let $u\in \mathrm{SH}(\Omega )$ be locally integrable. Let $\eta $ be a bounded nonnegative Borel function on $\Omega $ which vanishes outside a compact subset of $\Omega $ and equals one on an open set $V\subset \Omega $; for instance, $\eta =\mathbf1_ V$ for an open set $V\Subset \Omega $, or $\eta \in C_ c^\infty (\Omega )$ with values in $[0,1]$ and $\eta =1$ on $V$. Let $\eta \mu _ u$ denote the measure $E\mapsto \int _{E\cap \Omega }\eta \, \mathrm d\mu _ u$ on $\mathbb R^ d$. Then there is a smooth real harmonic function $h$ on $V$ such that
\[ u=E_ d*(\eta \mu _ u)+h\qquad \text{on }V. \]
This equality holds at every point of $V$, including the points where $u=-\infty $.
Proof.
If $\eta $ vanishes outside the compact set $C\subset \Omega $, then $\eta \mu _ u$ is a positive Borel measure on $\mathbb R^ d$ with support in $C$ and total mass at most $\sup \eta \cdot \mu _ u(C)<\infty $. Let $p=E_ d*(\eta \mu _ u)$. By proposition 3.4.11, $p$ is subharmonic and locally integrable on $\mathbb R^ d$, and $\Delta p=\eta \mu _ u$. If $\chi $ is a test function with $\operatorname{Supp}\chi \subset V$, then $\int \chi \eta \, \mathrm d\mu _ u=\int \chi \, \mathrm d\mu _ u$ because $\eta =1$ on $V$; hence $\Delta p=\Delta u$ on $V$. By Weyl's lemma (theorem 3.1.11), the real distribution $u-p$ is represented on $V$ by a smooth harmonic function $h$; since $\operatorname{Re}h$ also represents $u-p$, the uniqueness in theorem 3.1.11 shows that $h=\operatorname{Re}h$ is real. Then $p+h$ is a locally integrable subharmonic function on $V$ that agrees with $u$ almost everywhere on $V$ (proposition 2.5.3), and theorem 3.4.12(2), applied on $V$, gives $u=p+h$ at every point of $V$.
$\square$
Pome
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