Pome

Proposition 5.14.22EFFC20773E2E

Proposition 5.14.22.EFFC20773E2E Let $\Omega \subseteq \mathbb {C}^ n$ be a bounded domain. Let $\varphi ,\psi \in \mathrm{PSH}(\Omega )$ be locally bounded. Assume that

  1. 32392BCFEA58

    for each $z\in \partial \Omega $, we have

    \[ \varliminf _{\Omega \ni w\to z} \Bigl (\psi (w)-\varphi (w)\Bigr )\geq 0; \]
  2. 8FB9E09F9A4A

    $\varphi \leq \psi $ a.e. with respect to $(\mathrm{dd}^{\mathrm{c}}\psi )^ n$,

Then $\varphi \leq \psi $.

Proof. Apply the local Bedford--Taylor domination principle with $u=\varphi$ and $v=\psi$. Its hypotheses are exactly $\liminf_{\Omega\ni w\to z}(v-u)\geq0$ on the boundary and $u\leq v$ almost everywhere for $(\mathrm{dd}^{\mathrm c}v)^n$. Thus its conclusion is $\varphi\leq\psi$. The earlier proof uses bounded strict comparison sets and local boundedness; continuity on the boundary and a uniform bound on the whole domain are not required.

Cite this page

Comments

Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.

Read this page’s discussion on GitHubPost a comment on GitHub