Smooth regularization of plurisubharmonic functions (Q1277523)

A new regularization (smoothing) method for plurisubharmonic functions is proposed. Being applied to a plurisubharmonic function \(u\) in a pseudoconvex domain \(G\subset \mathbb{C}^n\), it produces a sequence of functions \(u_j\in \text{PSH} (G)\cap C^\infty(G)\) decreasing to \(u\). Furthermore, a function \(u\in \text{PSH} (\mathbb{C}^n)\) of finite order \(\rho>0\) can be approximated by a smooth plurisubharmonic function \(v\) such that \(| u-v|\) has almost logarithmic or power-law growth outside a small set \(E\subset \mathbb{C}^n\). In the latter case, \(|\text{grad } v|\) has power-law bounds, too. As a consequence, \(u\) is approximated by \(\log| f|\) with \(f\) an entire function, such that \[ \bigl| u(z)- \log| f(z)| \bigr|\leq C_\alpha(1+| z|)^\alpha, \quad z\not\in E, \] with any \(\alpha> \frac 45 \rho\).