Suppose that X is an analytic subvariety of a Stein manifold M and that varphi is a plurisubharmonic (psh) function on X which is dominated by a continuous psh exhaustion function u of M. Given any number c > 1, we show that varphi admits a psh extension to M which is dominated by cu on M. We use this result to prove that any omega-psh function on a subvariety of the complex projective space is the restriction of a global omega-psh function, where omega is the Fubini-Study Kahler form.
Coman, Dan; Guedj, Vincent; and Zeriahi, Ahmed, "Extension of Plurisubharmonic Functions with Growth Control" (2010). Mathematics Faculty Scholarship. 22.
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