PomeACA94F989A1ECorollary 5.30.3.ACA94F989A1E Let $U,V\subset \mathbb C^ n$ be domains, let $F:U\to V$ be biholomorphic, and let $u$ be bounded and plurisubharmonic on $U$. Then
tends locally uniformly to zero as $\delta \downarrow 0$.
Proof. On a compact subset, $F$ and its inverse have uniformly bounded first derivatives. Thus, for a fixed $A>1$, the inverse image under $F$ of the ball of radius $\delta $ about $F(z)$ lies between the balls about $z$ of radii $\delta /A$ and $A\delta $. Consequently, $S_\delta (u\circ F^{-1})(F(z))$ and $S_\delta u(z)$ both lie between $S_{\delta /A}u(z)$ and $S_{A\delta }u(z)$, and the first assertion of lemma 5.30.1 shows that their difference tends to zero uniformly. The second assertion of the lemma, applied to $u$ and to $u\circ F^{-1}$, compares each of the two convolutions with the corresponding supremum regularization. $\square$
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