Corollary 10.3.10E6419D4591F9

Proof. Choose a ball $B''$ in the same coordinates with $\overline B\subset B''$ and $\overline{B''}\subset B'$. Write $\theta =\mathrm{dd}^{\mathrm{c}}h$ on $B''$, with $h$ smooth (lemma 6.14.2), and apply corollary 7.7.5 to the locally bounded plurisubharmonic function $\varphi +h$ on $B''$. Subtracting $h$ from the result on $B''$ and keeping $\varphi $ outside $B''$ gives $\psi $, since the result equals $\varphi +h$ on $B''\setminus B$; here $(\theta +\mathrm{dd}^{\mathrm{c}}\psi )^ n=(\mathrm{dd}^{\mathrm{c}}(\psi +h))^ n$ is the Bedford–Taylor product of the bounded plurisubharmonic function $\psi +h$. If $\psi '$ is another function with the stated properties, then by upper semicontinuity $\varlimsup _{z\to \zeta ,\, z\in B}\psi '(z)\leq \psi '(\zeta )=\varphi (\zeta )$ for every $\zeta \in \partial B$. Hence $\psi '+h$ belongs to the family whose upper envelope is $\psi +h$ in corollary 7.7.5, and $\psi '\leq \psi $ on $B$. If $\psi '$ is continuous on $\overline B$, then its boundary values $\varphi |_{\partial B}$ are continuous, and $\psi '+h$ is the solution of the homogeneous Dirichlet problem with the continuous boundary values $(\varphi +h)|_{\partial B}$; by the uniqueness in theorem 7.7.4, any two such continuous functions coincide. $\square$