Smooth regularization of plurisubharmonic functions

R. S. Yulmukhametov

Matematičeskie zametki, Tome 62 (1997) no. 2, pp. 312-320

Résumé

We consider the problem of approximating a given plurisubharmonic function by smooth plurisubharmonic functions. We propose a new constructive approximation method that permits one to obtain more detailed information about the approximating functions. Thus a function $u\in\operatorname{PSH}(\mathbb C^n)$ having finite growth order can be approximated by smooth functions $v\in\operatorname{PSH}(\mathbb C^n)$ so that the difference $|v-u|$ has almost logarithmic growth (Theorem 2). It can also be approximated so that the difference $|v-u|$ has a power-law growth; in this case, however, power-law estimates on $|\operatorname{grad}v|$ appear (Theorem 3).