Proposition 1.3 . [0285]

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Proposition 1.3.

Let $X,\,\varphi,\,u$ be as in Theorem A with $M={\mathbb{C}}^{n}$ , and let $\{m_{j}\}$ , $\{\gamma_{j}\}$ be as above. There exists a psh function $\psi$ on ${\mathbb{C}}^{n}$ so that $\psi\,|_{{}_{X}}=\varphi$ and for all $z\in{\mathbb{C}}^{n}$ we have

\psi(z)<\left\{\begin{array}[]{ll}\gamma_{j}u(z),\;{\rm if}\;m_{j-1}<u(z)\leq m_{j},\;j\geq 2,\\ \gamma_{1}\max\{u(z),0\},\;{\rm if}\;u(z)\leq m_{1}.\end{array}\right.